i
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
ii
FOREWORD
13th Algebraic Hyperstructures and its Applications Conference (AHA2017),
organized by the International Algebraic Hyperstructures Association will take place from
24th July to the 27th July 2017 in Istanbul, a fascinating city built on two Continents, divided
by the Bosphorus Strait and this is one of the greatest cities in the world where you can see a
modern western city combined with a traditional eastern city, it’s a melting pot of many
civilizations and different people.
The series of International Conferences on Algebraic Hyperstructures and
Applications (AHA) aims at bringing together researchers and academics for the presentation
and discussion of novel theories and applications of Algebraic. The conference covers a broad
spectrum of topics related to Algebraic Hyperstructures.
AHA2017 provides an ideal academic platform for researchers and scientists to
present the latest research findings in mathematics. The conference aims to bring together
leading academic scientists, researchers and research scholars to exchange and share their
experiences and research results about mathematics and engineering studies.
We would like to thank to Yildiz Technical University for their invaluable supports.
We would also like to thank to all contributors to conference, especially to keynote speakers
who share their significant scientific knowledge with us, to organizing and scientific
committee for their great effort on evaluating the manuscripts. We do believe and hope that
each contributor will get benefit from the conference.
We hope to see you in 14th Algebraic Hyperstructures and its Applications
Conference (AHA2020) at Romania.
Yours Sincerely,
Prof. Dr. Bayram Ali ERSOY
Chair of AHA2017
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
iii
Previous AHA Events
1978, held in Taormina, Italy (November 25-28) Organized by P.Corsini
1983, held in Taormina, Italy (October 21-24) Organized by P.Corsini
1985, held in Udine, Italy (October, 15-18) Organized by P.Corsini
1990, held in Xanthi, Greece (June 27-30) Organized by T.Vougiouklis
1993, held in Iasi, Romania (July 4-10) Organized by M.Stefanescu
1996, held in Prague, Czech Republic (September 1-7) Organized by T.Kepca
1999, held in Taormina, Italy (June 13-19) Organized by R.Migliorato
2002, held in Samothrace, Greece (September 1-9) Organized by T.Vougiouklis
2005, held in Babolsar, Iran (September 1-7) Organized by R.Ameri
2008, held in Brno, South Moravia, Czech Republic (September 3-9) Organized by
S.Hoskova
2011, held in Pescara, Italy (October, 16-21) Organized by A. Maturo
2014, held in Xanthi, Greece (September, 2-7) Organized by S. Spartalis
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
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Conference Organization
Chair
Bayram Ali ERSOY (Turkey)
A. Göksel AĞARGÜN (Turkey)
Scientific Committee
· Piergiulio Corsini (Italy)
· Ivo Rosenbeg (Canada)
· Thomas Vougiouklis (Greece)
· Stephen Comer (U.S.A.)
· Violeta Leoreanu Fotea (Romania)
· Mohhamad Mahdi Zahedi (Iran)
· James Jantosciak (U.S.A)
· Reza Ameri (Iran)
· Stefanos Spartalis (Greece)
· Mirela Stefanescu (Romania)
· Tomas Kepka (Czech Republic)
· Renato Migliorato (Italy)
· Bijan Davvaz (Iran)
· Jan Chvalina (Czech Republic)
· Sarka Hoskova (Czech Republic)
· Antonio Maturo (Italy)
· Nemec Petr (Czech Republic)
· Maria Konstantinidou – Serafimidou
(Greece)
· R.A. Borzooei (Iran)
· Mario De Salvo (Italy)
· Christos Massouros (Greece)
· Mashhoor Refai (Jordan)
· Irina Cristea (Slovenia)
· Demetrious Stratigopoulos (Greece)
· Maria Scafati Tallini (Italy)
· A. R. Ashrafi, (Iran)
· Yuming Feng, (China)
· Rosaria Rota (Italy)
· Achilles Deamalides (Greece)
· Ian Tofan (Romania)
· Yupaporn Kemprasit (Thailand)
· Rita Procesi (Italy)
· Mohammad Reza Darafsheh (Iran)
· Aldo Ventre (Italy)
· Huakang Yang (China)
· Krassimir Atanassov (Bulgaria)
· Bayram Ali Ersoy (Turkey)
· K. P. Shum (China)
· Bal Kishan Dass (India)
· Tariq Mahmood (Pakistan)
· Antonios Kalampakas (Greece)
· Murat Sarı (Turkey)
· Nikolaos Antampoufis (Greece)
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
v
Conference organization Co-chair
Serkan ONAR (Turkey)
Local Organizing Committee
E. Mehmet Özkan (Turkey)
Adem Cengiz Çevikel (Turkey)
Murat Kirişci (Turkey)
İbrahim Demir (Turkey)
Filiz Kanbay (Turkey)
Ünsal Tekir (Turkey)
Mutlu Akar (Turkey)
Pınar Albayrak (Turkey)
Elif Demir (Turkey)
S. Ebru Daş (Turkey)
Murat Turhan (Turkey)
Fatma Çeliker (Turkey)
Ashraf Ahmed (Palestine)
Elif Segah Öztaş (Turkey)
M.Emin Köroğlu (Turkey)
Deniz Sönmez (Turkey)
Seda Akbıyık (Turkey)
Mücahit Akbıyık (Turkey)
Rabia Nagehan Üregen (Turkey)
Elif Aktaş (Turkey)
Melis Bolat (Turkey)
Sanem Yavuz (Turkey)
Emre Ersoy (Turkey)
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
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Contents
Foreword……………………………………………………………………………………….ii
Previous AHA Events……………………………………………………………………… ...iii
Conference Organization……………………………………………………………………...iv
Bijan Davvaz Some Applications of Algebraic Hyperstructures………………….1
Thomas Vougiouklis The Hv-matrix Representations…………………………………….4
Reza Ameri Some categorical aspects of algebraic hyperstructures……………..6
Piergiulio Corsini Hyperstructures and some of the most recent applications…………8
M. Mehdi Zahedi Weak Closure operations on ideals of a BCK-algebra……………...9
Lumnije Shehu Extensions of Polygroups by Polygroups via Factor Polygroups….10
Murat Alp On crossed polysquares and fundamental relations………………..12
Michal Novák Recent advances in EL-hyperstructures……………………….......15
Sinem Tarsuslu HH∗− Intuitionistic Heyting Valued Ω-Algebra…………………..16
Sinem Tarsuslu Algebraic Approach to Multiplicative Set…………………………17
Najmeh Jafarzadeh On the relation between categories of (m, n)-ary hypermodules
and (m, n)-ary modules……………………………………………19
Yıldıray Çelik Soft Bi-Ideals of Soft LA-Semigroups…………………………….23
Jan Chvalina Sequences of groups and hypergroups of linear ordinary
differential operators……………………………………………….24
Ümit Deniz On Different Approach of Fuzzy Ring Homomorphisms………....25
Mahmood Bakhshi L-hyperstructures………………………………………………….26
M. Golmohamadian New connections between hyperstructures and Graph Theory……31
Akbar Paad Ideals in HvMV-algebras………………………………………….33
Hashem Bordbar Overview on the Height of a Hyperideal in Krasner Hyperrings….37
Hashem Bordbar Theory of Double-framed soft set theory on Hyper BCK-algebra...39
Ioanna Iliou On P-hopes and P-Hv-structures on the plane……………………..40
Banu Pazar Varol On Neutrosophic Linear Spaces…………………………………...41
Nesibe Kesicioğlu The relationships between the orders induced by implications
and uninorms……………………………………………………...42
Nesibe Kesicioğlu A survey on order-equivalent uninorms…………………………...43
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
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Olga Cerbu The reflector functor and lattice L (R)…………………………….44
Deniz Sönmez A note on 2-absorbing δ-primary fuzzy ideals of commutative
Rings……………………………………………………………... 46
Rajab Ali Borzooei Relation Between Hyper EQ-algebras and Some Other
Hyper Structures…………………………………………………..47
Şerife Yılmaz Fuzzy hyperideals in ordered semihyperrings……………………..48
Şerife Yılmaz Fuzzy interior hyperideals in ordered semihyperrings…………….49
Ali Taghavi Hyperhilbert Spaces……………………………………………….50
Dilek Bayrak The Lattice Structure of Subhypergroups of a Hypergroup……….51
Tuğba Arkan Intuitionistic Fuzzy Weakly Prime Ideals…………………………52
Karim Ghadimi Some Results on Tensor Product of Krasner Hypervector Spaces...53
Jafar Azami Fuzzy coprimary submodules and their representation……………57
Güzide Şenel Constructing Topological Hyperspace with Soft Sets……………..58
Gülşah Yeşilkurt Fuzzy Weakly Prime Γ-ideals……………………………………...59
Sanem Yavuz Intuitionistic Fuzzy 2-absorbing Ideals of Commutative Rings…...60
Adem C. Cevikel Transition from Two-Person Zero-Sum Games to Cooperative
Games with Fuzzy Payoffs………………………………………..61
Karim Abbasi On computation of fundamental group of a finite hypergroup…….62
Hossein Shojaei Various kinds of quotient of a canonical hypergroup……………...64
Didem S. Uzay On multipliers of hyper BCC-algebras…………………………….66
Şule Ayar Özbal Derivations on hyperlattices……………………………………….67
Naser Zamani On fuzzy φ-prime ideals…………………………………………...68
Thawhat Changphas On pure hyperideals in ordered semihypergroups…………………69
Afagh Rezazadeh Relation Between Hyper K-algebras and Superlattice
(Hypersemilattice)………………………………………………….70
Tahere Nozari Vague Soft Hypermodules………………………………………....75
Zahra Soltani An introduction to Zero-Divisor Graph of a Commutative
Multiplicative Hyperring…………………………………………..76
Niovi Kehayopulu Some ordered hypersemigroups which enter their properties into
their σ-classes……………………………………………………..77
Serkan Onar Study of Γ-hyperrings by fuzzy hyperideals with respect to a
t-norm……………………………………………………………..78
Kostaq Hila On an algebra of fuzzy m-ary semihypergroups…………………..81
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
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Habib Harizavi On Annihilator in Pseudo BCI-algebras…………………………...83
Krisanthi Naka The embedding of an ordered semihypergroup in terms of
fuzzy sets………………………………………………………….84
Ashraf Abumghaiseeb On δ-Primary Hyperideals of Commutative Semihyperrings……..86
Mutlu Akar On the Vahlen Matrices…………………………………………...87
Abderrahmane Bouchair On the Baireness of function spaces………………………………88
Farida Belhannache Global asymptotic stability of a higher order difference equation..89
Kelaiaia Smail Weak alpha favorability of C(X) with a set open topology………90
Nemat Abazari Curves on Lightlike Cone in Minkowski Space…………………..91
Mehmet E. Köroğlu A class of LCD codes from group rings…………………………..92
Sümeyra Uçar New Blocking Cryptography Models……………………………..93
Nihal Taş A New Coding Theory with Generalized Pell (p,i) – Numbers…..94
Sezer Sorgun Some Problems in Spectral Graph Theory………………………..95
Hakan Küçük On trees which have exactly 4 non-zero Randi¢ eigenvalues…….96
Nurten Bayrak Gürses One-Parameter Planar Motions in Generalized Complex Number
Plane CJ…………………………………………………………...97
Esma Demir Çetin A New Approach to Motions and Surfaces with Zero Curvatures in
Lorentz 3-Space…………………………………………………100
Hülya Aytimur Chen-Ricci and Wintgen Inequalities for Statistical Submanifolds of
Quasi-Constant Curvature……………………………………….101
Mücahit Akbıyık One-Parameter Homothetic Motion on the Galilean Plane……...102
Murat Sarı Comparison of encryption and decryption algorithms through
various approaches………………………………………………103
Çağla Ramis Surfaces with Constant Slope and Tubular Surfaces…………….104
Ufuk Çelik New Contributions to Fixed-Circle Results on S-Metric Spaces..105
Hatice Tozak On The Parallel Ruled Surfaces With B-Darboux Frame……….106
Murat Kirişci A ANFIS Perspective for the diagnosis of type II diabetes……..108
Elif Demir Transitive Operator Algebras and Hyperinvariant Subspaces…..109
Süleyman Demir Reformulation of compressible fluid equations in terms of
Biquaternions……………………………………………………110
Erdal Gül On The Trace Formula for a Differential Operator of Second Order
with Unbounded Operator Coefficients…………………………111
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24-27 July, 2017, Istanbul-Turkey
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Mustafa Bayram Gücen Practical Stability Analyses of Nonlinear Fuzzy Dynamic Systems
of Unperturbed Systems with Initial Time Difference………….112
Sebahat Ebru Das A Numerical Scheme for Solving Nonlinear Fractional Differential
Equations in the Conformable-Derivative Sense………………..113
Fatma Öztürk Çeliker On invariant ideals on locally convex solid Riesz spaces………114
Pınar Albayrak On weakly compact-friendly operators…………………………115
Murat Turhan Hirota type discretization of Clebsch equations………………..116
Fatma Bulut An h-deformation of the superspace R(1|2) via a contraction….117
Mücahit Akbıyık Euler-Savary’s Formula On Dual Plane………………………...119
Ayten Özkan Mistakes and misconceptions regarding to natural numbers on
secondary Mathematics Education……………………………..121
Filiz Kanbay On a Fuzzy Application of the Particulate Matter Estimation….122
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yazd University, Department of Mathematics, Yazd, Iran
E-mail: [email protected]
Some Applications of Algebraic Hyperstructures
Bijan Davvaz 1
In this study, we describe the applications of algebraic hyperstructures and survey
related works. Hyperstructures represent a natural extension of algebraic structures and they
were introduced in 1934 by F. Marty. He generalized the notion of groups by defining
hypergroups. Algebraic hyperstructures have many applications in various sciences. In [1],
Corsini and Leoreanu presented some of the numerous applications of algebraic
hyperstructures, especially those from the last fifteen years, to the following subjects:
geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata,
cryptography, codes, median algebras, relation algebras, artificial intelligence and
probabilities. The largest class of hyperstructures is the one that satisfies weak axioms, i.e.,
the non-empty intersection replaces the equality. These are called Hv-structures and they were
introduced in 1990 by Vougiouklis [2]. The latter hyperstructures have many applications to
different disciplines like Biology, Chemistry, Physics, and so on. In several papers, Davvaz et
al. [3-9] introduced some chemical examples of hyperstructures. For instance, algebraic
hyperstructures associated to chain reactions; algebraic hyperstructures associated to
dismutation reactions; algebraic hyperstructures associated to redox reactions; hyperstructures
associated to electrochemical cells. Another motivation for the study of hyperstructures comes
from biology. In [10], the main objective of authors is to provide examples of hyperstructures
associated to inheritance. They explored the algebraic hyperstructure that naturally occurs as
genetic information gets passed down through generations. Mathematically, the algebraic
hyperstructures that arise in genetics are very interesting ones. They are generally
commutative and weakly associative. Moreover, many of the algebraic properties of these
hyperstructures have genetic significance. Indeed, there is an interplay between the purely
algebraic hyperstructures and the corresponding genetic properties, that makes the subject so
fascinating. The examples given in [10] indicated that the theory of genetic hyperstructure
algebras is generally worth practicing. Mendel, the father of genetics took the first steps in
defining “contrasting characters, genotypes in F1 and F2 . . . and setting different laws”. The
1
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yazd University, Department of Mathematics, Yazd, Iran
E-mail: [email protected]
genotypes of F2 is dependent on the type of its parents genotype and it follows certain roles.
In [11], the authors analyzed the second generation genotypes of monohybrid and a dihybrid
with a mathematical structure. They used the concept of Hv-semigroup structure in the F2-
genotypes with cross operation and proved that this is an Hv-semigroup. They also
determined the kinds of number of the Hv-subsemigroups of F2-genotypes. In [12], the
authors provided examples about different types of inheritance (Mendelian and Non-
Mendelian inheritance) and relate them to hyperstructures and generalize the work done in
[10]. The feature of hyperstructures allows us to extend this theory into the elementary
particle physics. In [13], the authors have considered one important group of the elementary
particles, Leptons. They have shown this set that along with the interactions between its
members can be described by the algebraic hyperstructure. In [14], Asghari-Larimia and
Davvaz presented a connection between algebraic hyperstructures and number theory. They
introduced a hyperoperation associated to the set of all arithmetic functions and analyzed the
properties of this new hyperoperation. Several characterization theorems are obtained,
especially in connection with multiplicative functions. Then, Al Tahan and Davvaz [15]
constructed a hyperring structure on the set of arithmetic functions.
Keywords: hyperstructure, chemistry, biology, physics, number theory.
2010 AMS Classification: 20N20
References:
1. Corsini, P. and Leoreanu, V., Applications of hyperstructures theory, Advances in
Mathematics, Kluwer Academic Publisher, 2003.
2. Vougiouklis, T., The fundamental relation in hyperrings. The general hyperfield,
Algebraic hyperstructures and applications (Xanthi, 1990), 203-211, World Sci. Publishing,
Teaneck, NJ, 1991.
3. Davvaz, B., Dehghan Nezad, A. and Benvidi, A., Chain reactions as experimental
examples of ternary algebraic hyperstructures, MATCH Communications in Mathematical
and in Computer Chemistry, 65(2), 491-499, 2011.
4. Davvaz, B., Dehghan Nezhad, A. and Benvidi, A., Chemical hyperalgebra: Dismutation
reactions, MATCH Communications in Mathematical and in Computer Chemistry, 67,
55- 63, 2012.
5. Davvaz, B. and Dehghan Nezhad, A., Dismutation reactions as experimental
verifications of ternary algebraic hyperstructures, MATCH Communications in
2
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yazd University, Department of Mathematics, Yazd, Iran
E-mail: [email protected]
Mathematical and in Computer Chemistry, 68, 551-559, 2012.
6. Davvaz, B., Dehghan Nezad, A. and Mazloum-Ardakani, M., Chemical hyperalgebra:
Redox reactions, MATCH Communications in Mathematical and in Computer Chemistry,
71, 323-331, 2014.
7. Davvaz, B., Dehghan Nezad, A., Mazloum-Ardakani, M. and Sheikh-Mohseneib, M.A.,
Describing the algebraic hyperstructure of all elements in radiolytic processes in cement
medium, MATCH Communications in Mathematical and in Computer Chemistry, 72,
375-388, 2014.
8. Davvaz, B., Weak algebraic hyperstructures as a model for interpretation of chemical
reactions, Iranian Journal of Mathematical Chemistry, 7(2), 267-283, 2016.
9. Al Tahan, M. and Davvaz, B., Weak chemical hyperstructures associated to
electrochemical cells, Iranian Journal of Mathematical Chemistry, to appear.
10. Davvaz, B., Dehghan Nezad, A. and Heidari, M.M., Inheritance examples of algebraic
hyperstructures, Information Sciences, 224, 180-187, 2013.
11. Ghadiri, M., Davvaz, B. and Nekouian, R., Hv-Semigroup structure on F2-offspring of a
gene pool, International Journal of Biomathematics, 5(4), 1250011 (13 pages), 2012.
12. Al Tahan, M. and Davvaz, B., Hyperstructures associated to biological inheritance,
Mathematical Biosciences, 285, 112-118, 2017.
13. Dehghan Nezhad, A., Moosavi Nejad, S.M., Nadjafikhah and Davvaz, B., A physical
example of algebraic hyperstructures: Leptons, Indian Journal of Physics, 86(11),
1027-1032, 2012.
14. Asghari-Larimi, M. and Davvaz, B., Hyperstructures associated to arithmetic functions,
ARS Combinatoria, 97, 51-63, 2010.
15. Al Tahan, M. and Davvaz, B., On the existence of hyperrings associated to arithmetic
functions, Journal of Number Theory, 174, 136-149, 2017.
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Democritus University of Thrace, School of Education 681 00 Alexandroupolis, Greece
E-mail: [email protected]
The Hv-matrix Representations
Thomas Vougiouklis1
Emeritus Professor
The Theory of Representations of Hyperstructures was started in mid 80’s but that
time there was not any general definition of hyperfield. The Hv-structures, were introduced in
4th AHA Congress 1990, and at the same time, the general definition of the hyperfield, was
given. Since then the Theory of Representations is refereed mainly on Hv-groups by Hv-
matrices, that is that, the matrices have entries elements of an Hv-field or from an Hv-ring. In
Hv-structures the weak axioms replace the classical axioms of structures by replacing the
‘equality’ by the ‘non empty intersection’. The characteristic property of Hv-structures, is that
a partial order on Hv-structures on the same underline set, is defined. The weak properties
increase extremely the number of hyperstructures defined in the same set, therefore it is
reasonable to find applications in mathematics and in other applied sciences, as well. On the
other side, in order to obtain strict results, we ask from applied sciences to give more axioms
and more restrictions. This is the case, for example, in nuclear physics with Santilli’s iso-
theory. In representation theory the researchers have to treat well almost all the classical
algebraic structures from semigroups to Lie-algebras. We present the problems, some new
results and we give to researchers open problems in mathematics from hyperstructures.
Keywords: Hyperstructures, Hv-structures, Hv-matrix.
2010 AMS Classification: 20N20.
References:
1. Corsini P., Leoreanu V., Application of Hyperstructure Theory, Klower Acad. Publ., 2003.
2. Davvaz B., Leoreanu V., Hyperring Theory and Applications, Int. Academic Press, 2007.
3. Vougiouklis T., The fundamental relation in hyperrings. The general hyperfield, 4thAHA,
Xanthi 1990, World Scientific, (1991), 203-211.
4. Vougiouklis T., Hyperstructures and their Representations, Monographs in Math.,
Hadronic, 1994.
4
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Democritus University of Thrace, School of Education 681 00 Alexandroupolis, Greece
E-mail: [email protected]
5. Vougiouklis T., On Hv-rings and Hv-representations, Discrete Math., Elsevier, 208/209,
1999, 615-620.
6. Vougiouklis T., Finite Hv-structures and their representations, Rend. Sem. Mat. Messina
S.II, V.9, 2003, 245-265.
7.Vougiouklis T., Hypermathematics, Hv-structures, hypernumbers, hypermatrices and Lie-
Santilli admissibility, American J. Modern Physics,4(5), 2015, 34-46.
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, , Department of Mathematics, University of Tehran, Tehran, Iran.
E-mail: [email protected]
Some Categorical Aspects of Algebraic Hyperstructures
Reza Ameri1
In this study we briefly discuses on some of algebraic hyperstructures theory in view
point of category theory and present some features of the various hyperstructures such as
hypergroups, hyperrings, hypermodules and etc. In this regards we investigate various
categories of hyperstructures based on various kinds of morphisms, especially on multivalued
homomorphisms. We will proceed by introducing some categorical objects such as, zero
object, product, coproduct and free objects. Finally, we constructs some functors from
categories of hyperstructures to the correspondence classical algebraic category.
Today hyperstructures rapidly developed in view point of theory and application and
many concept of classical algebra are appear in this theory. In parallel to this progress main
questions will rise about and some terminology has been used improper. On the other hands,
some of terminology may be bad used. Also, for study the relationships between
hyperstructures and classical algebra we need to use the exact language to correct
mathematically descriptions of these notions. For example in hyperstructures theory we use
the phrases such as: a hypergroup is a generalization of a group, the class of polygroup is a
generalization of ordinary group. The fundamental relation on a hypergroup is a function
which assign to each hypergroup a group or in general to every hyperalgebra one can assign
an algebra via the fundamental relation. Here naturally give rise some main questions:
What is different between the class of hypergroups and groups?
Are they really mathematically different?
And many other questions which appears to study of algebraic hyperstructures and its
relationship to the related algebraic structures. In this paper we briefly to mention the role of
category theory as a useful tools to answer to these questions as well as we introduce some
categorical objects such as product, kernel, free and etc. in category of Krasner hypermodules.
Keywords: category, hypergroups, hypermodules, fundamental functor, hyperadditive
category.
2010 AMS Classification: 03G99, 06B99, 06F05.
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, , Department of Mathematics, University of Tehran, Tehran, Iran.
E-mail: [email protected]
Acknowledgements:
The author partially has been supported by "Algebraic Hyperstructure Excellence (AHETM),
Tarbiat Modares University, Tehran, Iran" and "Research Center in Algebraic hyperstructures
and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran".
References:
1. R. Ameri, On the categories of hypergroups and hypermodules, J. Discrete Math. Sci. cryptogr. 6
(2003) 121-132.
2. R. Ameri, M.Norouzi, Prime and primary hyperideales in Krasner (m; n)-hyperring, European J.
Combin. 34(2013)379-390.
3. R. Ameri, M.Norouzi, V. Leoreanu-Fotea, On Prime and primary subhypermodules of (m; n)-
hypermodules, European J. Combin. 44(2015)175-190.
4. SM. Anvariyeh, S. Mirvakili, B. Davvaz, Fundamental relation on (m; n)- hypermodules over (m;
n)-hyperrings, Ars combin. 94(2010)273-288.
5. Z. Belali, SM. Anvariyeh, S. Mirvakili, B. Free and cyclic (m; n)-hypermodules, Tamkang J.
Math.42(2011) 105-118.
6 P. Corsini, Prolegemena of Hypergroup Theory, second ed. Aviani, Editor, 1993.
7. P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, in: Advances in
Mathematices, Vol. 5, Kluwer Academic Publishers, 2003.
8. G. Crombez, On (m; n)-rings, Abh. Math. Sem. Univ. Hamburg 37(1972)180-199.
9. G. Crombez, J. Timm, On (m;n)-quotient rings, Abh. Math. Sem. Univ. Hamburg,37(1972)200-203.
10. B. Davvaz, V. Leoreanu, Hyperring Theory and Applacations, International Academic Press, 2007,
p. 8.
11. B. Davvaz, T. Vougiouklis , n-ary hypergroups, Iran.J. Sci. Technol. Trans. A. Sci. 30(A2)(2006)
165-174.
12. W. Dornte, Untersuchungen Uber einen verallgemeinerten Gruppenenbegri,Math. Z. 29(1928) 1-
19.
13. V. Leoreanu, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math. 24(2008).
14. V. Leoreanu-Fotea, B. Davvaz, n-hypergroups and binary relations, European J. Combin. 29(2008)
1027-1218.
15. V. Leoreanu-Fotea, B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci. 178(2008) 4114-
4124.
16. F. Marty, Sur une generalization de group in: 8iem congres des Mathematiciens Scandinaves,
Stockholm. 1934, pp. 45-49.
17. S. Mirvakili, B. Davvaz, Constructions of (m; n)-hyperrings. MATEMAT. 67,1(2015)1-16.
18. S. Mirvakili, B. Davvaz, Relations on Krasner (m; n)-hyperrings. European J. Combin. 31(2010)
790-802.
19. H. Shojaei, R. Ameri, Some Results On Categories of Krasner Hypermodules, J. Fundam. Appl.
Sci. 2016, 8(3S), 2298-2306.
20. T. Vougiouklis, Hyperstructure and their Representations, Hardonic, Press, Inc, 1994.
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics and Computer Sciences, University of Udine, Udine, Italia
E-mail: [email protected]
Hyperstructures and some of the most recent applications
Piergiulio Corsini1
After a brief history of Hypergroups, since the beginning around the 40s till
today, one gives an excursus of the most recent applications of this topic to Fuzzy Sets and
Chinese groups as HX-hypergroups.
Keywords: Hyperstructures
2010 AMS Classification: 20N20
References:
1. P.Corsini, Prolegomena of Hypergroup Theory, Aviani Editore (1993), pp. 216.
2. P.Corsini, Algebra per Ingegneria, Aviani Editore, (1991)
3. P.Corsini, Introducere in Theoria Hipergrupurilor, Translated by V.Leoreanu, Editura
Universitatii “Al. I. Cuza” Iasi, (1998) pp.158
4. P.Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Advances in
Mathematics, vol. 5, Kluwer Academic Publishers, (2003).
8
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Shahid Bahonar University, Kerman, Iran ,
2Department of Mathematics, Graduate University of Advanced Technology, Mahan-Kerman, Iran,
E-mail(s): [email protected]
Weak Closure operations on ideals of a BCK-algebra
Hashem Bordbar1 and Mohammad Mehdi Zahedi
2
Weak closure operation, which is more general form than closure operation,on ideals
of BCK-algebras is introduced, and related properties are investigated. Regarding weak
closure operation, finite type, (strong) quasi-primeness, tender and naive are considered.
Using a weak closure operation “cl” and an ideal A of a lower BCK-semilattice X with the
greatest element 1, a new ideal K of X containing the ideal Acl of X is established. Using this
ideal K, a new function
clt : I(X) → I(X); A →K
is given, and related properties are considered. We show that if “cl” is a tender (resp., naive)
weak closure operation on I(X), then so are “clt” and “clf ”.
Keywords: closure operation, (finite type, tender, naive) weak closure operation,
zeromeet element, meet ideal.
2010 AMS Classification: 06F35, 03G25.
References:
[1] H. Bordbar and M. M. Zahedi, Semi-prime closure operations on BCK-
algebra, Commun. Korean Math. Soc. 30 (2015), no. 5, 385–402.
[2] H. Bordbar and M. M. Zahedi, A finite type of closure operations on
BCK-algebra, Appl. Math. Inf. Sci. Lett. 4 (2016), no. 2, 1–9.
[3] Y. Huang, BCI-algebra, Science Press, Beijing 2006.
[4] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. 1994.
9
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan 2Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan
E-mails:[email protected], [email protected]
EXTENSION OF POLYGROUPS BY POLYGROUPS VIA FACTOR POLYGROUPS
Lumnije Shehu1, Hani Khashan
2
The idea of constructing extensions of polygroups via factor polygroups comes from
an extension that De Salvo introduced in [13] which is called ( , )H G -hypergroups.
Basically, given a hypergroup ( , )H and mutually disjoint sets { }i i GA where G is a given
group such that 0A H . Set ii G
K A
and define a hyper operation on K as follows: For
all ,x y H , x y x y . For all ix A and jy A such that i jA A H H , kx y A
where i j k . This extension of G by H represents a hypergroup. The wreath product
[ ]H G introduced in [2] can be obtained by De Salvo’s construction when H and G are
polygroups, 0A H and { }iA i for 0i . In our construction, we consider two polygroups
H and L . We restrict the cardinalities of sets iA , 0i to be equal to the cardinality of
some factor polygroup /H I and the cardinality of 0A equals to that of H . The hyper
operation on ii L
K A
is based on the hyper operations on the factor polygroup /H I and the
polygroup L . In principle, the element zero of L is enlarged by the polygroup H and the
rest of the elements of L are enlarged by isomorphic copies of the factor polygroup /H I .
This construction yields a polygroup in the case when the subpolygroup I is normal.
However, the kernel of a strong homomorphism is not necessarily normal, [10]. Therefore, by
weakening the condition of normality, we obtain the utmost possible extensions. Indeed, we
define and study regularly normal subpolygroups. After introducing the isomorphism
theorems subject to these subpolygroups, we are able to present our new extension via factor
polygroups.
Keywords: hypergroups, polygroups, polygroups extensions, regularly normal
subpolygroups.
2010 AMS Classification: 20N20
References:
[1] M. Alp, B. Davvaz, Crossed polymodules and fundamental relations, U.P.B. Sci. Bull.,
Series A, 77(2): 129-140, 2015.
10
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan 2Department of Mathematics, Al al-Bayt University, Al Mafraq, Jordan
E-mails:[email protected], [email protected]
[2] S. D. Comer, Extension of polygroups by polygroups and their representations using
colour schemes, Lecture notes in Math., 1004: 91-103, 1982.
[3] S. D. Comer, A remark on chromatic polygroups, Congr. Numer., 38: 85-95, 1983.
[4] S. D. Comer, Constructions of color schemes, Acta Univ. Carolin. Math. Phys., 24: 39-48,
1983.
[5] S. D. Comer, Some problems on hypergroups, Fourth Int. Con. on AHA, 67-74, 1990.
[6] S. D. Comer, Combinatorial aspects of relations, Algebra Universalis., 18: 77-94, 1984.
[7] P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani Editore, 1993.
[8] P. Corsini, V. Loreanu, Application of Hyperstructure Theory, Kluwer: Academic
Publishers, 2003. [9] B. Davvaz, On polygroups and permutation polygroups, Math.
Balkanica (N.S.), 14: 41-58, 2000. [10] B. Davvaz, Isomorphism theorems of polygroups,
Bull. Malays. Math. Sci. Soc., 33(2): 385-392, 2010.
[11] B. Davvaz, Polygroup Theory And Related Systems, World Scientific Publishing Co.,
2013.
[12] B. Davvaz, V. Leoreanu-Fotea, Hyperring theory and applications, International
Academic Press, Palm Harbor, Fla, USA, 2007.
[13] M. De Salvo, Gli (H,G)-ipergruppi, Riv. Mat. Univ. Parma, 10: 207-216, 1984.
[14] M. De Salvo, G. Lo Faro, On the n*-complete hypergroups, Discrete Mathematics,
208/209: 177-188, 1999. 177-188.
[15] M. Dresher and O. Ore, Theory of Multigroups, Amer. J. Math., 60: 705-733, 1938.
[16] J. Jantosciak, Homomorphisms, equivalences and reductions in hypergroups, Riv. Mat.
Pura Appl., 9: 23-47, 1991.
[17] C. G. Massouros, Some properties of certain subhypergroups, Ratio Mathematica, 25:
67-76, 2013.
[18] M. Tallini, Hypergroups and geometric spaces, Ratio Mathematica, 22: 69-84, 2012.
[19] T. Vougiouklis, Hv-groups defined on the same set, Discrete Math., 155: 259-265, 1996.
11
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Mohammad Ali Dehghani Department of Mathematics, Yazd University, Yazd, Iran 2Bijan Davvaz Department of Mathematics, Yazd University, Yazd,Iran 3Murat Alp Department of Mathematics, Nigde Ömer Halisdemir University, Nigde, Turkey
E-mail(s): [email protected], [email protected], [email protected]
On crossed polysquares and fundamental relations
Mohammad Ali Dehghani1, Bijan Davvaz
2, Murat Alp
3
In this paper, we introduce the notion of crossed polysquare of polygroups and we give some
of its properties. Our results extend the classical results of crossed squares to crossed
polysquares. One of the main tools in the study to polygroups is the fundamental relations.
These relations connevt polygroups to groups, and on the other hand, introduce new important
classes. So, we consider a crossed polysquare and by using the concept of fundamental
relation, we obtain a crossed square.
Keywords: Crossed module, crossed square, polygroup, fundamental relation.
2010 AMS Classification: 13D99, 20N20, 18D35
References:
1- M. Alp, Actor of crossed modules of algebroids, Proc. 16th Int. Conf. Jangjeon Math.
soc.,16(2005) 6-15.
2- M. Alp, Pullback crossed modules of algebroids, Iranian J. Sci. Tech.,Transaction A,
32(A3) (2008) 145-181.
3- M.~Alp, Pullbacks of profinite crossde modules and cat 1-profinite groups, Algebras
Groups Geom, 25 (2) (2008) 215-221.
4- M. Alp and B. Davvaz, On Crossed Polymodules and Fundamental Relations, U.P.B. Sci.,
Bull., Series A, 77 (2) (2015) 129-140.
5- M. Alp and Ö. Gürmen, Pushouts of profinite crossed modules and cat 1-profinite groups,
Turkish Journal of Mathematics, 27 (2003) 539-548.
6- Z. Arvasi , Crossed squares and 2-crossed modules of commutative algebras, Theory and
Applications of Categories, 3 (7) (1997) 160-181.
7- Z. Arvasi and T. porter, Freeness conditions for 2-crossed modules of commutative
algebras, Applied Categorical Structures, (to appear).
8- Z. Arvasi and E. Ulualan, On algebraic models for homotopy 3-types, Journal of
Homotopy and Related Structures, 1 (1) (2006) 1-27.
9- Z. Arvasi and E. Ulualan, 3-Types of simplicial groups and braided regular crossed
modules, Homotopy and Applications, 9 (1) (2007) 139-161.
10- H. J. Baues, Combinatorial Homotopy and 4-Dimensional Compexes, Walter de Gruyter,
Berlin, De Gruyter expositions in Mathematics, (1991).
11- R. Brown , Computing Homotopy types using crossed N-cubes of groups, Adams
Memorial Symposium on Algebraic Topology, 1 (1992) 187-210.
12- R. Brown and N. D. Gilbert, Algebraic models of 3-types and automorphism structures
12
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Mohammad Ali Dehghani Department of Mathematics, Yazd University, Yazd, Iran 2Bijan Davvaz Department of Mathematics, Yazd University, Yazd,Iran 3Murat Alp Department of Mathematics, Nigde Ömer Halisdemir University, Nigde, Turkey
E-mail(s): [email protected], [email protected], [email protected]
for crossed modules, Proc. London Math. Soc. 59 (3) (1989) 51-73.
13- R. Brown and J. L. Loday, Van Kampen theorems for diagrams of spaces, Topology, 26
(1987) 311-334.
14- R. Brown and G. H. Mosa, Double categories, R-categories and crossed modules, U. C.
N. W maths preprint 88 (11). (1988) 1-18.
15- P. Carrasco , A. M. Cegarra and A. R. Garzòn, The classifying space of categorical
crossed module, Mathematische Nachrichaten, 283 (4). (2010) 544-567.
16- S. D. Comer, Polygroups derived from cogroups, J. Algebra, 89 (1984) 397-405.
17- D. Conduchè, Modules crois\ès gènèralisès de longueur 2, J. Pure Applied Algebra 34
(1984) 155-178.
18- D. Conduchè, Simplicial crossed modules and mapping cones, Georgian Math. J. 10 (4)
(2003) 623-636.
19- P. Corsini, Prolegomena of hypergroup theory, Second edition, Ariain editor(1993).
20- B.Davvaz, A survey on polygroups and their properties,Proceedings of the International
Confrence on Algebra, (2010) 148-156, Sci. Publ., Hackensack, NJ, (2012).
21- B.Davvaz, Applications of the $r^*$-relationPolygroup theory and related systems, World
Sci. Publ., (2013).
22- B. Davvaz, On Polygroups and Permutation Polygroups, Math., Balkanica (N. S.),14 (1-2)
(2000) 41-58.
23- B. Davvaz, Isomorphism theorems of polygroups, Bulletin of the Malaysian Mathematical
Sciences Society (2), 33 (3) (2010) 385-392.
24- B. Davvaz, Polygroup theory and related systems, World Sci. Publ., 2013.
25- D.Freni, A note on the core of a hypergroup and the transitive closure 𝛽∗ of 𝛽, Riv. Math.
Pura Appl., 8 (1991) 153-156.
26- D. Guin-Walery and J. L. Loday, Obstructionà 1'excision en k-thèories algèbrique, In
Friedlander, E. M., Stein, M. R. (eds.)Evanston conf. On Algebraic k-Theory (1980). (Lect.
Notes Math. 854) Springer, Berlin, Heidelberg, New York (1981) 179-216.
27- F. J. Korkes and J. Porter, Profinite crossed modules,U. C. N. W pure mathematics
preprint 86 (11). (1986).
28- M. Koskas, Groupoids, demi-groups et hypergroups, J. Math. Pures Appl., 49 (1970)
155-192.
29- V. Leoreanu-Fotea, The heart of some important classes of hypergroups, Pure Math.
Appl., 9 (1998)351-360.
30- J. L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Appl.~ Algebra
24 (1982) 179-202.
31- K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France,
(1990) 118.,129-146.
32- J. Porter, N-Types of Simplicial Groups And Crossed N-Cubes, Topology,~32 (1).(1993)
5-24.
13
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Mohammad Ali Dehghani Department of Mathematics, Yazd University, Yazd, Iran 2Bijan Davvaz Department of Mathematics, Yazd University, Yazd,Iran 3Murat Alp Department of Mathematics, Nigde Ömer Halisdemir University, Nigde, Turkey
E-mail(s): [email protected], [email protected], [email protected]
33- T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Inc, 115, Palm
Harber, USA (1994).
34- J. H. C.Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949) 453-
496.
14
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech Republic 2Masaryk University, Faculty of Economics and Administration, Brno, Czech Republic
E-mail(s): [email protected], [email protected],
Recent Advances in EL-hyperstructures
Michal Novák1, Štěpán Křehlík
2
EL-hyperstructures are semihypergroups or ring-like hyperstructures S constructed
from partially (or, in many cases, quasi-) ordered semigroups, where the hyperoperation on S
is defined by 𝑎 ∗ 𝑏 = [𝑎 . 𝑏)≤ = {𝑥 ∈ 𝑆 |𝑎 . 𝑏 ≤ 𝑥} for all 𝑎, 𝑏 ∈ 𝑆 . When looking for
examples, one can construct numerous EL-semihypergroups, hypergroups, join spaces,
lattice-like or ring-like hyperstructures in a number of natural contexts as the set S can be a
number domain, set of words of a given alphabet, set of objects, properties of which can be
described by numbers, sets of vectors or matrices, etc. The relations can be numerous as well:
ordering numbers (or numerically described properties) by size, divisibility relation, or
relations motivated by some special contexts. EL-hyperstructures were introduced by
Chvalina in [1] and named so and studied by Novák in e.g. [2,3].
In our paper we focus on some recent advances in the area of EL-hyperstructures. We
clarify the issue of antisymmetry of the relation ``≤" and include examples when it is a quasi-
ordering which is moreover symmetric, i.e. an equivalence. We show the use of the
construction in the area of lattice-like hyperstructures, i.e. for 𝐻𝑣 -semilattices,
hypersemilattices or hyperlattices (which were studied in [4]). We discuss implications of
extensivity of the hyperoperation, i.e. contexts when {𝑎, 𝑏} is included in 𝑎 ∗ 𝑏 for all 𝑎, 𝑏 ∈
𝑆 . We also mention the way EL-hyperstructures can be used to construct Cartesian
composition of multiautomata. Finally, we briefly mention the relation of EL-hyperstructures
to some other concepts of hyperstructure theory, where the idea of ordering is used, such as
ordered hyperstructures, quasi-order hypergroups or some special cases of BCI-algebras.
Keyword(s): EL-hyperstructures, hyperstructure theory, ordered semigroups, quasi-ordered
semigroup
2010 AMS Classification: 20N20, 06F05
Reference(s):
1. Chvalina J., Functional Graphs, Quasi-ordered Sets and Commutative Hypergroups,
Masaryk University, Brno, 1995. (in Czech)
2. Novák M., Some basic properties of EL-hyperstructures, European J.~Combin., 34, 446-
459, 2013.
3. Novák M., On EL-semihypergroups, European J. Combin., 44(Part B), 274-286, 2015.
4. Křehlík Š., Novák M., From lattices to 𝐻𝑣-matrices, An. St. Univ. Ovidius Constanta,
24(3), 209-222, 2016.
15
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Mersin University, Department of Mathematics, Mersin, Turkey
E-mails:[email protected], [email protected]
HH*-Intuitionistic Heyting Valued -Algebra
Sinem Tarsuslu(Yılmaz)1, Gökhan Çuvalcıoğlu
1
Intuitionistic Logic was introduced by L. E. J. Brouwer and Heyting algebra was
defined by A. Heyting in 1930, to formalize the Brouwer’s intuitionistic logic. The concept of
Heyting algebra has been accepted as the basis for intuitionistic propositional logic. Heyting
algebras have had applications in different areas. The co-Heyting algebra is the same lattice
with dual operation of Heyting algebra. Also, co-Heyting algebras have several applications
in different areas.
In this paper, we introduced the new concept HH*- Intuitionistic Heyting Valued -
Algebra. The purpose of introducing this new concept is to expand the field of researchers’
area using both membership degree and non-membership degree. This allows us to get more
sensitive results.The concepts of HH*- Intuitionistic Heyting valued set, HH*- Intuitionistic
Heyting valued relation, HH*- Intuitionistic Heyting valued -algebra and the
homomorphism over HH*- Intuitionistic Heyting valued -algebra were defined.
Keyword(s): Heyting Valued Algebra, co-Heyting Valued Algebra, Omega Algebra,
Intuitionistic Logic.
2010 AMS Classification: 03C05
Reference(s):
1. Brouwer, L. E. J., Intuitionism and Formalism, English translation by A. Dresden, Bulletin
of the American Mathematical Society, 20 (1913): 81--96, reprinted in Benacerraf and
Putnam (eds.) 1983: 77--89; also reprinted in Heyting (ed.) 1975: 123--138.
2. Çuvalcığlu G., Heyting Valued Omega Free Algebra, Çukurova University Institute of
Science and Technology, PhD Thesis, Adana, 2002, 68 p.
3. Faith C., Algebra: Rings, Modules and Categories I, Springer-Verlag, Berlin
4. Heyting, A. "Die formalen Regeln der intuitionistischen Logik," in three parts,
Sitzungsberichte der preussischen Akademie der Wissenschaften: 42--71, 158--169, 1930.
English translation of Part I in Mancosu 1998: 311--327.
5. Lawvere F.V. , Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes, in
A.Carboni, et.al., Category theory, Proccedings, Como. 1990.
This study was supported by the Research Fund of Mersin University in Turkey with Project Number:
2015-TP3-1249.
16
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Mersin University, Department of Mathematics, Mersin, Turkey
E-mail(s):[email protected], [email protected]
Algebraic Approach to Multiplicative Set
Sinem Tarsuslu(Yılmaz)1, Gökhan Çuvalcıoğlu
1
Rough set theory was introduced by Pawlak in 1982. The theory of rough sets is an
extension of set theory as a subset of a universe is defined by a pair of ordinary sets called the
lower and upper approximations.The algebraic structures of rough sets were studied by
several authors.Davvaz introduced the notion of rough subring in 2004. Some properties of
the lower and the upper approximations in a ring were examined by Davvaz.
In this study, we examined some relations between rough sets and multiplicative
subsets of a commutative ring R. The lower and upper approximations of the set X with
respect to " " were defined and algebraic properties were examined.
Keyword(s): Rough set,Rough subring, Multiplicative set, Fuzzy ideal.
2010 AMS Classification: Primary 05C38, 15A15; Secondary 05A15, 15A18
References:
1. Bonikowaski Z., Algebraic structures of rough sets, in: W.P.Ziarko (Ed.), Rough Sets
Fuzzy Sets, and Knowledge Discovery, springer-Verlag, Berlin, (1995), pp.242-247.
2. Biswas R., Nanda S., Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42
(1994), 251-254.
3. Corsini P., Rough sets, fuzzy sets and join spaces, Honorary volume dedicated to Prof.
Emeritus J.Mittas, Aristotle Univ. of Thessaloniki, (1999-2000).
4. Davvaz B., Roughness based on fuzzy ideals, Inform. Sci.176 (2006), 2417-2437.
5. Davvaz B., Rough sets in a fundamental ring, Bull. Iranian Math. Soc. 24 (1998), 49-61.
6. Dubois D., Prade H., Rough fuzzy sets and fuzzy rough sets, Int. J.General Syst. 17 (2-3)
(1990), 191-209.
7. Iwinski T., Algebraic approach to rough sets, Bull. Polish Acad. Sci. Math. 35 (1987), 673-
683.
8. Kuroki N., Rough ideals in semigroups, Inform. Sci. 100 (1997), 139-163.
9. Kuroki N., Mordeson J.N., Structure of rough sets and rough groups, J.Fuzzy Math. 5 (1)
(1997), 183-191.
10. Kuroki N., Wang P.P., The lower and upper approximations in a fuzzy group, Inform. Sci.
90 (1996) 203-220.
11. Mordeson J.N., Rough set theory applied to (fuzzy) ideal theory, Fuzzy Sets and Systems
121 (2001), 315-324.
12. Pawlak Z., Rough sets, Int. J.Comput. Inform. Sci. 11 (1982), 341-356.
17
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Mersin University, Department of Mathematics, Mersin, Turkey
E-mail(s):[email protected], [email protected]
13. Pawlak Z., Rough Sets-Theoretical Aspects of Reasoning about Data, Kluwer Academic
Publishers, Dordrecht, (1991).
14. Pawlak Z., Skowron A., Rough sets: some extensions, Inform. Sci. 177 (1) (2007), 28-40.
15. Pomykala J., Pomykala J.A., The stone algebra of rough sets, Bull. Polish Acad. Sci.
Math. 36 (1988) 495-508.
16. Sarkar M., Rough-fuzzy functions in classification, Fuzzy Sets Syst. 132 (2002) 353--369.
17. W. Liu, Operations on fuzzy ideals, Fuzzy Sets and Systems 8(1983), 31-41.
18. Zadeh L.A., Fuzzy Sets, Information and Control, 8, (1965), p. 338-353.
19. Zadeh L.A., The concept of linguistic variable and its applications to approximate
reasoning,
Part I, Inform. Sci. 8 (1975) 199-249;
Part II, Inform. Sci. 8 (1975) 301-357;
Part II, Inform. Sci. 9 (1976) 43-80;
This study was supported by the Research Fund of Mersin University in Turkey with Project Number:
2015-TP3-1249.
18
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
(1) Department of Mathematics, Payamnour University. P.O. Box 19395-3697, Tehran, Iran.
(2) School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran,
P.O. Box 14155-6455, Tehran, Iran.
[email protected], [email protected]
On the relation between categories of ),( nm -ary hypermodules and ),( nm -ary modules
N .Jafarzadeh(1)
,R. Ameri(2)
We introduce the category of ),( nmR -hypermodules over a Krasner ),( nm -hyperring R and
obtain some categorical objects in this category such as product and coproduct. We apply the
fundamental relations * and
* on, M and, R respectively to construct fundamental functor from
the category of ),( nmR -hypermodules into category of */R -modules. In particular we consider the
fundamental relation on ),( nm -hypermodules, and construct functor from the category of ),( nm -
hypermodules to the category of ),( nm -modules. Then, we find the relations between hom, product,
coproduct and fundamental functor.
In this section we give some definitions and results of n-ary hyperstructures which we need in
what follows.
A mapping 𝑓:𝐻 × 𝐻 × …× 𝐻⏟ 𝑛
→ 𝑃∗(𝐻) is called an n-ary hyperoperation, where 𝑃∗(𝐻)is the
set of all non-empty subsets of H.
Definition 2.3 [11] A Krasner ),( nm -hyperring is algebraic hyperstructure ),,( khR
which satisfies the following axioms:
• ),( hR is a canonical m -ary hypergroup;
• ),( kR is an n -ary semigroup;
• the n -ary operation k is distributive to the m -ary hyperoperation ,h i.e, for all
,,, 11
1
1 Rxaa mn
i
i
and ,1 ni
));,,(,),,,((=)),(,( 1
1
111
1
111
1
1
n
im
in
i
in
i
mi axakaxakhaxhak
• 0 is a zero element (absorbing element), of the n -ary operation ,., eik for Rxn 2 we
hav:e
,0).(==),0,(=)(0, 2322
nnn xkxxkxk
Definition 2.5 [3] A Krasner ),( nm -hypermodule ),,( gfM is an ),( nm -hypermodule
with a canonical m -ary hypergroup ),( fM over a Krasner ),( nm -hyperring ).,,( khR
Definition 2.8 [11] Let ),,( khR be ),( nm -hyperring.The relation * is the smallest
equivalence relation such that the quotient )/,/,/( *** khR be ),( nm -ring. where */R is the set
of equivalence classes. The * is called fundamental equivalence relation.
19
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
(1) Department of Mathematics, Payamnour University. P.O. Box 19395-3697, Tehran, Iran.
(2) School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran,
P.O. Box 14155-6455, Tehran, Iran.
[email protected], [email protected]
Various categories of (m, n) -ary hypermodules
Definition 3.2 The category HmodR nm ),( of ),( nm -ary hypermodules defined as follows:
• the objects of HmodR nm ),( are ),( nm - hypermodules,
• for the objects M and ,K the set of all morphisms from M to K is defined as follows:
},smhomomorphiman is )(:|{=),( * KPMffKMHomR
• the composition gf of morphisms )(: * KPMf and )(: * LPKg defined as
follows:
),(=)( ),(:)(
* tgxgfKPHgfxft
• for any object ,H the morphism ),(:1 * HPHH defined by },{=)(1 xxH is the
identity morphism.
Theorem 3.5 modRhmodR nmnms *
),(),( /:F defined by */=)( MMF and
,=)( *F is a functor ,//: and : *
2
*
1
*
21 MMMM where modR nm *
),( / is the
category of all ),( nm -modules over ./ *R
Remark 3.9 In the following of this paper we consider the category of all ),( nm -
hypermodules over a ),( nm -hyperring ,R in the sense of Krasner ),( nm -hypermodules over
commutative Krasner ),( nm -hyperring R with identity. We denote this category by
.),( KHmodR nm Hence, the objects of KHmodR nm ),( are Krasner ),( nm -hypermodules over
commutative Krasner ),( nm -hyperring with identity and all morphisms are multivalued
homomorphisms.
In this section, concepts of direct hyper product and direct hyper coproduct of a Krasner
),( nm -hypermodule are defined. Also we give some properties of the category .),( KHmodR nm
Definition 4.3 Let }|{ IiM i be a family of ),( nm -hypermodules. we define a
hyperoperation on i
Ii
M
as follows:
.}{ )(|}{=}{ 111
i
Ii
im
i
im
iiii
im
i MaafttaF
20
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
(1) Department of Mathematics, Payamnour University. P.O. Box 19395-3697, Tehran, Iran.
(2) School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran,
P.O. Box 14155-6455, Tehran, Iran.
[email protected], [email protected]
For Rr and ,i
Ii
i Ma
define
.),(=)}{()(
1)(
1)(
1)(
1 Iii
n
iIii
n argarG
then ,i
Ii
M
together with m -ary hyperoperation F and n -ary operation G is called direct hyper
product }.|{ IiM i
Definition 4.6 The direct hyper sum of the family }|{ IiM i of ),( nm -hypermodules,
denoted by i
Ii
M
is the set of all ,}{ Iiia where ia can be non-zero only for a finite number of
indices.
Theorem 4.11 Let }|{ IiM i be a family of ),( nm -hypermodules over an ),( nm -ary
hyperring R and let Iii
M ,* and )( **
iM
Ii
iM
Ii
be fundamental equivalence relation on iM and
)( i
Ii
i
Ii
MM
respectively. then
• ,/)/(: **
1i
Mi
IiiM
Ii
i
Ii
MM
• ./)/(: **
2i
Mi
IiiM
Ii
i
Ii
MM
Theorem 4.12 Fundamental functor F preserves zero object, product and coproduct.
Proposition 5.3 Let }|{ IiM i be a family of ),( nm -hypermodules over an ),( nm -ary
hyperring R and N also is an ),( nm -hypermodule and F be fundamental functor . Then
))).,((()),(( NMhomNMhom iR
Ii
i
Ii
R FF
Corollary 5.5 Let CBA ,, be ),( nm -hypermodules over an ),( nm -ary hyperring R and
and F be fundamental functor . Then isomorphism
)),(()),(()),(( CBhomCAhomCBAhom RRR FFF
is natural.
21
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
(1) Department of Mathematics, Payamnour University. P.O. Box 19395-3697, Tehran, Iran.
(2) School of Mathematics, Statistics and Computer Science, Collage of Sciences, University of Tehran,
P.O. Box 14155-6455, Tehran, Iran.
[email protected], [email protected]
Keywords: category, ),( nm -hypermodules, product, coproduct, additive category.
References
[1] R. Ameri, On the categories of hypergroups and hypermodules, J. Discrete Math. Sci.
cryptogr. 6 (2003) 121-132.
[2] R. Ameri, M.Norouzi, Prime and primary hyperideales in Krasner ),( nm -hyperring,
European J. Combin. 34(2013)379-390.
[3] SM. Anvariyeh, S. Mirvakili, B. Davvaz, Fundamental relation on ),( nm -hypermodules
over ),( nm -hyperrings. Ars combin. 94(2010)273-288.
[4] P. Corsini, Prolegemena of Hypergroup Theory, second ed. Aviani, Editor, 1993.
[5] B. Davvaz, V. Leoreanu, Hyperring Theory and Applacations, International Academic
Press, 2007, p. 8.
[6] B. Davvaz, T. Vougiouklis , n -ary hypergroups, Iran.J. Sci. Technol. Trans. A. Sci.
30(A2)(2006) [7] W. Dörnte, Untersuchungen Über einen verallgemeinerten Gruppenenbegriff, Math.
Z. 29(1928) 1-19.
[8] V. Leoreanu, Canonical n -ary hypergroups, Ital.J. Pure Appl. Math. 24(2008).
[9] F. Marty, Sur une generalization de group in: iem8 congres des Mathematiciens
Scandinaves, Stockholm. 1934, pp. 45-49.
[10] S. Mirvakili, B. Davvaz, Constructions of ),( nm -hyperrings. MATEMAT.
67,1(2015)1-16.
[11] S. Mirvakili, B. Davvaz, Relations on Krasner ),( nm -hyperrings. European J. Combin.
31(2010) 790-802.
[12] H. Shojaei, R. Ameri, Some Results On Categories of Krasner Hypermodules. J
FundamAppl Sci. 2016, 8(3S), 2298-2306.
[13] T. Vougiouklis, Hyperstructure and their Representations, Hardonic, Press, Inc, 1994.
22
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Ordu University, Department of Mathematics, Ordu, Turkey
E-mails:[email protected]
Soft Bi-Ideals of Soft LA-Semigroups
Yıldıray Çelik1
In this paper, we present notion of soft bi-ideal of a soft ring and give some results on
it. Also, we introduce concept of soft bi-ideal of a soft LA-semigroup and investigate some
properties of it.
Keywords: soft ring, soft bi-ideal, soft LA-semigroup
2010 AMS Classification: 06F05, 16D25
Reference(s):
1. Acar U., Koyuncu F., Tanay B., Soft sets and soft rings, Comput. Math. Appl., 59, 3458-
3463, 2010.
2. Aktaş H., Çağman N., Soft sets and soft groups, Inform. Sci., 177, 2726-2735, 2007.
3. Ali M.I., Shabir M., Naz M., Algebraic structures of soft sets associated with new
operations, Comput. Math. Appl., 61(9), 2647-2654, 2011.
4. Aslam M., Shabir M., Mehmood A., Some studies in soft LA-semigroups, J. Adv. Res.
Pure Math., 3(4), 128-150, 2011.
5. Çelik Y., Ekiz C., Yamak S., A new view on soft rings, Hacet. J. Math. Stat., 40(2), 273-
286, 2011.
6. Feng F., Jun Y.B., Zhao X., Soft semirings, Comput. Math. Appl., 56, 2621-2628, 2008.
7. Maji P.K., Biswas R., Roy A.R., Soft set theory, Comput. Math. Appl., 45, 555-562, 2003.
8. Molodtsov D., Soft set theory-first results, Comput. Math. Appl., 37, 19-31, 1999.
23
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech 2 Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech 3 Brno University of Technology, Faculty of Electrical Engineering and Communication, Brno, Czech
E-mail(s):[email protected], [email protected], [email protected]
Sequences of Groups and Hypergroups of Linear Ordinary Differential Operators
Jan Chvalina1, Michal Novák
2, David Staněk
3
Linear ODE's are a classical tool for constructing many useful models for description
of numerous processes. Given their standard forms, left-hand sides of such equations (both of
homoegeneous and non-homogeneous) are called linear differential operators. Groups of such
operators of different orders can be constructed and some of their properties studied. This
includes solvability, relation to quasi-automata or actions on specific spaces or structures.
Using a suitable ordering or quasi-ordering of groups of linear differential operators
we construct hyperstructures of linear differential operators. Then, with the help of these
hyperstructures, we construct multiautomata which are cardinal sums of perfect semisimple
submultiautomata.
The main objective of our paper is to focus on the study of sequences (finite or
countable) of groups and hypergroups of linear differential operators of decreasing orders. For
this we use inclusion embeddings and group homomorphisms. In particular, we obtain and
study what we call ``coupled sequences". We also include a construction of sequences of
second-order linear differential operators in the Jacobi form, i.e. such operators that the
coefficient at the first-order derivative is zero. Our results can be generalized to operators of
an arbitrary order.
We also apply our considerations to the theory of (multi-)automata. In particular, we
obtain actions of abelian groups and hypergroups constructed from linear spaces of
polynomials of various dimensions over additive abelian groups of differential operators of
the corresponding order with constant coefficients at the highest-order derivatives.
Keyword(s): hyperstructure theory, linear differential operators, ODE, theory of automata
2010 AMS Classification: 20N20, 68Q70, 47D03
Reference(s):
1. Bavel Z., The source as a tool in automata, Information and Control, 18, 140-155, 1971.
2. Jan J., Digital Signal Filtering, Analysis and Restoration, IEEE Publ. London, 2000.
3. Gécseg F., Péak I., Algebraic Theory of Automata, Budapest, Akadémia Kiadó, 1972.
4. Chvalina J., Novák M. and Křehlík Š., Cartesian composition and the problem of
generalising the MAC condition to quasi-multiautomata. An. St. Univ. Ovidius Constanta,
24(3), 79-100, 2016
5. Chvalina J., Staněk, D. Sequences of automata formed by groups of polynomials and by
semigroups of linear differential operators, in: 16th Conference on Applied Mathematics
APLIMAT 2017. Proceedings. SUT in Bratislava, 2017.
24
On Different Approach of Fuzzy Ring Homomorphims (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Recep Tayyip Erdogan University, Department of Mathematics, Rize, Turkey
E-mail(s):[email protected]
On Different Approach of Fuzzy Ring Homomorphims
Umit DENİZ1
In this study we approach the definition of TLring homomorphism. In literature the
definition of fuzzy ring homomorphism is given by using the classic functions. In this study
we give the definition of fuzzy ring homomorphism by using the definition of Mustafa
Demirci’s fuzzy function. Some definition and theorems of ring homomorphism in classic
algebra is adapted to fuzzy algebra and proved.
Keywords: Fuzzy Functions, Fuzzy Equivalence Relations, Triangular Norms, Fuzzy
Subrings, Fuzzy Ideals
2010 AMS Classification: Mathematics
References:
1. Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.
2. Rosenfeld A., Fuzzy groups, J.Math Anal Appl., 35 , 512-517, 1971.
25
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Bojnord, Iran E-mail: [email protected]
𝓛-hyperstructures
Mahmood Bakhshi1
In this paper, as a generalization of familiar classical ordered algebraic structures such
as ordered semigroups and ordered groups the notion of 𝓛 -hyperstructure is introduced.
Giving some examples it is shown that the familiar ordered hyperstructures and also those
hyper algebraic structures arose from logic can be viewed as a special types of 𝓛 -
hyperstructres. After that investigating basic properties, some types of hyperideals are
introduced, thier properties are investigated and some characterizations and the connections
among them are obtained.
Keyword(s): hyperstructure, ordered sets, algebras of logics
2010 AMS Classification: 20N20, 06F15, 06F35, 06D35
1. Main results
Definition 2.1. By a language of hyperstructures we mean a set 𝓛 consists of a set ℛ of
relation symbols and a set ℱ of set-valued function symbols such that to each member of ℛ (
of ℱ) is associated a natural number (a non-negative integer) called the arity of the symbol.
ℱ𝑛 denotes the set of set-valued function symbols in ℱ of arity 𝑛, and ℛ𝑛 denotes the set of
relation symbols in ℛ of arity 𝑛.
Definition 2.2. Let 𝓛 be a language. An ordered pair 𝑨 =< 𝐴; 𝓛 > in which 𝐴 is a non-empty
set and 𝓛 consists of a family 𝑅 of fundamental relations 𝑟𝑨 on 𝐴 indexed by ℛ and a family
ℱ of fundamental hyper operations 𝑓𝑨 on 𝐴 indexed by ℱ is called a hyperstructure of type 𝓛
(or 𝓛 -hyperstructure). 𝐴 is called the universe of 𝑨. When ℛ = ∅, 𝑨 is a hyper algebra and if
ℱ = ∅, 𝑨 is a relational structure. If 𝓛 is finite, say ℱ = {𝑓1, … , 𝑓𝑚} and ℛ = {𝑟1, … , 𝑟𝑛}, we
often write < 𝐴; 𝑓1, … , 𝑓𝑚; 𝑟1, … , 𝑟𝑛 > instead of < 𝐴; 𝓛 >. If 𝑓𝑖 ∈ ℱ𝑛 is an 𝑛𝑖-ary function
symbol and 𝑟𝑗 ∈ ℛ𝑙 is an 𝑙𝑗-ary relation we write, for brevity, ℒ<𝑛1,…,𝑛𝑚;𝑙1,…,𝑙𝑘>-hyperstructure,
where 𝑛1 ≥ 𝑛2 ≥ ⋯ ≥ 𝑛𝑚 and 𝑙1 ≥ 𝑙2 ≥ ⋯ ≥ 𝑙𝑘, instead of 𝓛 -structure. İn this paper, we
focus on ℒ<2;2>-hyperstructures; for brevity, whenever it is clear from the context, we write
ℒ-hyperstructure instead of ℒ<2;2>-hyperstructure.
Definition 2.3. Let < 𝐻;∘, ≤> be an ℒ-hyperstructure. The relation ≤ can be eneralized to
nonempty subsets of 𝐻 as follows: for 𝐴, 𝐵 ∈ 𝑃∗(𝐻),
(i) 𝐴 ≤𝑤 𝐵, if there exist 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 such that 𝑎 ≤ 𝑏,
(ii) 𝐴 ≤𝑟𝑤 𝐵, if for each 𝑏 ∈ 𝐵 there exists 𝑎 ∈ 𝐴 such that 𝑎 ≤ 𝑏,
26
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Bojnord, Iran E-mail: [email protected]
(iii) 𝐴 ≤𝑙𝑤 𝐵, if for each 𝑎 ∈ 𝐴 there exists 𝑏 ∈ 𝐵 such that 𝑎 ≤ 𝑏,
(iv) 𝐴 ≤𝑡𝑤 𝐵, if 𝐴 ≤𝑙𝑤 𝐵 and 𝐴 ≤𝑟𝑤 𝐵,
(v) 𝐴 ≤𝑠 𝐵 if for each 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵 we have 𝑎 ≤ 𝑏.
Definition 2.4. Let < 𝐻;∘, ≤ > be an ℒ-hyperstructure. We say that ≤ is
(i) weak left (right) compatible if
𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑤 𝑎 ∘ 𝑦 ( 𝑥 ∘ 𝑎 ≤𝑤 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻
If ≤ is weak left and weak right compatible it is said to be weak compatible.
(ii) r-left (r-right) compatible if
𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑟𝑤 𝑎 ∘ 𝑦 ( 𝑥 ∘ 𝑎 ≤𝑟𝑤 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻
If ≤ is r-left and r-right compatible it is said to be r-compatible.
(iii) l-left (l-right) compatible if
𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑙𝑤 𝑎 ∘ 𝑦 (𝑥 ∘ 𝑎 ≤𝑙𝑤 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻
If ≤ is l-left and l-right compatible it is said to be l-compatible.
(iv) t-left (t-right) compatible if ≤ is l-left and r-left compatible (l-right and r-right
compatible).
If ≤ is t-left and t-right compatible it is said to be t-compatible or briefly compatible.
(v) strong left (right) compatible if
𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑥 ≤𝑠 𝑎 ∘ 𝑦 (𝑥 ∘ 𝑎 ≤𝑠 𝑦 ∘ 𝑎) ∀𝑎 ∈ 𝐻
If ≤ is strong left and strong right compatible it is said to be strong compatible.
Definition 2.5. Let ≤ be any types of the relations introduced in Definition 2.3. We say that ≤
is reversed left (right) compatible if
𝑥 ≤ 𝑦 ⟹ 𝑎 ∘ 𝑦 ≤ 𝑎 ∘ 𝑥 (𝑟𝑒𝑠𝑝. 𝑦 ∘ 𝑎 ≤ 𝑥 ∘ 𝑎) ∀𝑎 ∈ 𝐻
Definition 2.6. By a (weak, l-, r-, t-, strong) ℒ<2;2>-hyperstructure we mean an ℒ-
hyperstructure on which is defined a (weak, l-, r-, t-, strong) compatible binary relation.
Remark 2.7. For convenience, we drop the prefix two-sided and so a two-sided ℒ-
hyperstructure is called an ℒ-hyperstructure.
Definition 2.8. An ℒ-hyperstructure < 𝐻;∘, ≤ > in which ∘ is commutative (associative) is
said to be a commutative ℒ-hyperstructure (resp, ℒ-semihypergroup).
Definition 2.9. An element e of an ℒ-hyperstructure < 𝐻;∘, ≤ > is called an identity if
𝑥 ∈ 𝑥 ∘ 𝑒 ∩ 𝑒 ∘ 𝑥, ∀𝑥 ∈ 𝐻.
Example 2.10.
(i) Any hyper 𝐾-algebra [] and any hyper 𝑀𝑉-algebra [] is an ℒ<2;2>-hyperstructure in
which the binary relation satisfies Definition 2.3(i).
27
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Bojnord, Iran E-mail: [email protected]
(ii) Any hyper residuated lattice [] is an ℒ<2;2>-hyperstructure in which the binary
relation is weak right compatible with respect to the multiplication and weak left
compatible with respect to the residuation.
(iii) Any hyper 𝐵𝐶𝐾-algebra [] is an ℒ<2;2>-hyperstructure in which the binary relation is
reversed 𝑙-left compatible.
(iv) Any ordered semihypergroup [] is an ℒ<2;2>-hyperstructure with an 𝑙-left compatible
relation.
(v) Consider ℝ1 = [1, ∞), the set of all real numbers greater than 1, as a poset with the
natural ordering, and define 𝑥 ∘ 𝑦 to be the set of all upper bounds of {𝑥, 𝑦}. Thus
< ℝ1;∘, ≤ >, is a commutative r-ℒ<2;2>-semihypergroup with 1 as the unique
identity.
(vi) Let < 𝐺; ∗, 𝑒, ≤ > be an ordered group, and let 𝑥 ∘ 𝑦 =< {𝑥, 𝑦} >, the subgroup of 𝐺
generated by {𝑥, 𝑦}. Then < 𝐺; ∘, ≤ > is a commutative ℒ<2;2>-hyperstructure.
(vii) Let < 𝐿; ∨, ∧, 0 > be a lattice with the least element 0. For 𝑎, 𝑏 ∈ 𝐿, let 𝑎 ∘ 𝑏 =
𝐹(𝑎 ∧ 𝑏), where 𝐹(𝑥) is the principal filter generated by 𝑥 ∈ 𝐿. Then, < 𝐿; ∘ > is a
commutative r- ℒ-hyperstructure.
(viii) Let 𝐻 = {𝑎, 𝑏} be a chain with 𝑎 < 𝑏. We define a hyperoperation `∘' on 𝐻 as in
Table 1. Then, < 𝐻; ∘, ≤ > is an r- ℒ -hyperstructure, whereas it is not l- ℒ -
hyperstructure because 𝑎 ≤ 𝑏 but 𝑎 ∘ 𝑎 ≰𝑙𝑤 𝑎 ∘ 𝑏. Indeed, 𝑏 ∈ 𝑎 ∘ 𝑎 but there is not
any element 𝑥 ∈ 𝑎 ∘ 𝑏 such that 𝑏 ≤ 𝑥.
Table 1: Cayley table of Example 2.10(viii)
∘ 𝒂 𝒃
𝒂 {𝑎, 𝑏} {𝑎}
𝒃 {𝑎} {𝑎, 𝑏}
Definition 2.11. By an ordered ℒ -hyperstructure we mean an ℒ -hyperstructure in which the
binary relation is a partial ordering.
Definition 2.14. Let 𝐻 be a (weak, left, right, strong) ordered ℒ-hyperstructure. A down set 𝐼
of 𝐻 is called a
(i) left hyperideal if 𝐻𝐼 ⊆ 𝐼,
(ii) right hyperideal if 𝐼𝐻 ⊆ 𝐼.
If 𝐴 is a left and a right hyperideal, it is called a hyperideal of 𝐻.
Example 2.15.
(i) Consider the ordered ℒ -hyperstructure < 𝕫,∘, ≤ > given in Example 2.10(vi). It is
not difficult to check that the only hyper ideal of 𝕫 is itself.
28
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Bojnord, Iran E-mail: [email protected]
(ii) Let 𝐻 = {𝑎, 𝑏, 𝑐} be a partially ordered set, where 𝑎 < 𝑏 and define
hyperoperation `∘' on 𝐻 as shown in Table 2. Then < 𝐻,∘, ≤ > is an ordered ℒ-
semihypergroup in which 𝐼 = {𝑎, 𝑏} is a hyperideal of 𝐻.
Table 2: Cayley table of Example 2.15(ii)
∘ a b c
a {𝑎, 𝑏} {𝑎} {𝑎}
b {𝑎} {𝑏} {𝑏}
c {𝑎} {𝑏} {𝑐}
(iii) Consider the partially ordered set 𝐻 given in part (ii). We define a hyperoperation
on 𝐻 as in Table 3. Then < 𝐻,∘, ≤ > is an ordered ℒ -semihypergroup in which
𝐼 = {𝑎, 𝑏} is a left hyperideal of 𝐻 but it is not a right hyperideal because 𝑎 ∘ 𝑐 =
{𝑎, 𝑏, 𝑐} ⊈ 𝐼. Thus 𝐼𝐻 ⊈ 𝐼$.
Table 3: Cayley table of Example 2.15(iii)
∘ a b c
a {𝑎, 𝑏} {𝑎} {𝑎, 𝑏, 𝑐}
b {𝑎} {𝑏} {𝑐}
c {𝑎, 𝑏} {𝑎, 𝑏} {𝑎, 𝑏, 𝑐}
(iv) Consider the partially ordered set 𝐻 given in part (ii), again. Then, < 𝐻,∘, ≤ > is
an ordered ℒ -semihypergroup in which the hyperoperation is given as in Table 4.
It is easy to check that 𝐼 = {𝑎, 𝑏} is a right hyperideal of 𝐻 which is not a left
hyperideal becuase 𝑐 ∘ 𝑏 = {𝑐} ⊈ 𝐼. Hence, 𝐻𝐼 ⊈ 𝐼.
(v)
Table 4: Cayley table of Example 2.15(iv)
∘ a b c
a {𝑎, 𝑏} {𝑎} {𝑎, 𝑏}
b {𝑎} {𝑏} {𝑎, 𝑏}
c {𝑎, 𝑏, 𝑐} {𝑐} {𝑎, 𝑏, 𝑐}
Reference(s):
1. Ameri R., Bakhshi M., Nematollah Zadeh S. A., Borzooei R.A., Fuzzy (strong) congruence
relations on hypergroupoids and hyper BCK-algebras, Quasigroups Related Systems, 15, 11-
24, 2007.
2. Blyth T. S., Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
3. Borzooei R. A., Hasankhani A., Zahedi M. M., On hyper K-algebras, Math. Japonica 52,
113-121, 2000.
4. Corsini P., Prolegomena of hypergroup theory, 2nd edition, Aviani editor, 1993.
5. Ghorbani S., Hasankhani A., Eslami E., Hyper MV-algebras, Set-Valued Mathematics and
Applications, 1, 205-222, 2008.
29
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Bojnord, Iran E-mail: [email protected]
6. Heidari D., Davvaz B., On ordered hyperstructures, U.P.B. Sci. Bull. Series A, 73 (2), 85-
96, 2000.
7. Jun Y. B., Zahedi M. M., Xin X. L., Borzooei R. A., On hyper BCK-algebras, Italian
Journal of Pure and Applied Mathematics, 10, 127-136, 2000.
8. Marty F., Sur une generalization de la notion de group, 8th congress Math. Scandenaves,
Stockholm, 45-49, 1934.
9. Zahiri O., Borzooei R. A., Bakhshi M., (Quotient) hyper residuated lattices, Quasigroups
Related Systems 20, 125-138, 2012
30
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran 2 Faculty of Sciences and Modern Technologies, Graduate University of Advanced Technology, Kerman, Iran
E-mail(s): [email protected], [email protected]
New connections between hyperstructures and Graph Theory
Masoumeh Golmohamadian 1 , Mohammad Mehdi Zahedi 2
In present study, we investigate the relation of dominating sets in graphs and
hyperstructures. We introduce different hyperoperations and semihypergroups, deriving from
dominating sets and minimal dominating sets of a graph and we examine their properties.
A vertex 𝑣 in a graph 𝐺 is said to dominate itself and each of its neighbors and a set 𝐷
of vertices of 𝐺 is a dominating set of 𝐺 if every vertex of 𝐺 is dominated by at least one
vertex of 𝐷 . let 𝐺 = (𝑉, 𝐸) be a graph, 𝐻𝑖 be a dominating set and 𝐻 be the set of all
dominating sets of 𝐺. Then we define 𝜃(𝐻𝑖) as the maximum number of vertices of 𝐻𝑖, that
we can omit from 𝐻𝑖 to convert it to a minimal dominating set. For every 𝐻𝑖 , 𝐻𝑗 ∈ 𝐻, we
define the commutative semihypergroup (𝐻,∗) in the following way:
𝐻𝑖 ∗ 𝐻𝑗 =
{
𝐻𝑖 , 𝑖𝑓 𝜃(𝐻𝑖) < 𝜃(𝐻𝑗)
𝐻𝑗 , 𝑖𝑓 𝜃(𝐻𝑗) < 𝜃(𝐻𝑖)
𝐻𝑖 , 𝑖𝑓 𝜃(𝐻𝑖) = 𝜃(𝐻𝑗) 𝑎𝑛𝑑 |𝐻𝑖| < |𝐻𝑗|
𝐻𝑗 , 𝑖𝑓 𝜃(𝐻𝑖) = 𝜃(𝐻𝑗) 𝑎𝑛𝑑 |𝐻𝑗| < |𝐻𝑖|
{𝐻𝑖 , 𝐻𝑗} , 𝑖𝑓 𝜃(𝐻𝑖) = 𝜃(𝐻𝑗) 𝑎𝑛𝑑 |𝐻𝑖| = |𝐻𝑗|
In addition, we construct another commutative semihypergroup (𝐻, 𝑜) by considering
𝜆(𝐻𝑖) as the set of all minimal dominating sets which have minimum cardinality among all
minimal dominating sets that are obtained from 𝐻𝑖.
We also make a connection between minimal dominating sets and hyperstructures. let
𝑆 be the set of all minimal dominating sets and 𝑆𝑖 be a minimal dominating set. Then we
define 𝜑(𝑆𝑖) by
𝜑(𝑆𝑖) = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 𝑜𝑓 𝐺[𝑆𝑖] ∕ |𝑆𝑖|
We introduce the commutative semihypergroup (𝑆,∗𝑖) as follows:
for every 𝑆𝑚, 𝑆𝑛 ∈ 𝑆
𝑆𝑚 ∗𝑖 𝑆𝑛 = {
𝑆𝑛 , 𝑖𝑓 𝜑(𝑆𝑚) < 𝜑(𝑆𝑛) 𝑆𝑚 , 𝑖𝑓 𝜑(𝑆𝑚) > 𝜑(𝑆𝑛){𝑆𝑚, 𝑆𝑛} , 𝑖𝑓 𝜑(𝑆𝑚) = 𝜑(𝑆𝑛)
We investigate some situations in which this semihypergroup is hypergroup and give some
examples to clarify them. Finally, we present a new class of graphs in which this
semihypergroup will be a hypergroup.
Keyword(s): Dominating sets in graphs: Minimal dominating sets in graphs:
Semihypergroup: Hypergroup
2010 AMS Classification: 05C69, 20N20
31
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran 2 Faculty of Sciences and Modern Technologies, Graduate University of Advanced Technology, Kerman, Iran
E-mail(s): [email protected], [email protected]
References:
1. Chartrand G., Lesnaik L., Graphs & Digraphs, Chapman & Hall, 1996.
2. Corsini P, Leoreanu V., Applications of Hyperstructure Theory, Kluwer Academic
Publishers, 2003.
3. Kalampakas A., Spartalis S., Path hypergroupoids: Commutativity and graph connectivity,
European Journal of Combinatorics, 44, 257–264, 2015.
4. Kalampakas A., Spartalis S., Tsigkas A., The Path Hyperoperation, Analele Stiintifice ale
Universitatii Ovidius Constanta, Seria Matematica, 22, 141-15, 2014.
5. Rosenberg I., Hypergroups induced by paths of a directed graph, Italian Journal of Pure
and Applied Mathematics, 4, 133-142, 1998.
32
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, University of Bojnord, Bojnord, Iran
[email protected], [email protected], [email protected]
Ideals in HvMV-algebras
Mahmood Bakhshi 1 , RoghayehTaherpoor 1 and Akbar Paad 1
In this paper, first some basic definitions are reviewed. Then some types of ideals such
as Hv MV-ideals, weak Hv MV-ideals and nodal Hv MV-ideals are introduced and some
characterizations and their properties are obtained.
Introduction
In 1958, Chang [3], introduced the concept of an MV-algebra as an algebraic proof of
the completeness theorem for ℵ0-valued Łukasiewicz propositional calculus. After that many
mathematicians haveworked on MV-algebras and obtained significant results. Mundici [7]
proved that MV-algebras and AbelianA-groups with strong unit are categoricallyequivalent.
After that Marty[6]introduced the notion of a hypergroup several authors worked on
hypergroups, especially in France and in the United States, but also in Italy, Russia and Japan.
Bakhshi et al. introduced ordered polygroups [2] which are subclasses of hypergroups on
which is defined a partial ordering with special property. Hyperstructureshave many
applications to several sectors of both pure and applied sciences. A short review of the theory
of hyperstructures appear in [4]. Vougiouklis [8] introduced a generalization of the well-
known algebraic hyperstructures such as hypergroup so-called Hv-structures. Actually some
axioms concerning the above hyperstructures such as the associative law, the distributive law
and so on are replaced by their corresponding weak axioms. In order to obtain a suitable
generalization of MV-algebras which may be equivalent (categorically) to a certain subclass
of the class of Hv -groups, the author introduced the concept of an Hv MV-algebra [1] and
obtained some related results.
HvMV-algebras: Basic properties
In this section, the concept of an HvMV-algebra is introduced. For more details we
refer to thereferences.
Deftnition 2.1.An Hv MV-algebra is a nonempty set H endowed with a binary hyperoperation
‘⊕’, a unary operation ‘∗’ and a constant ‘0’ satisfying the followingconditions:
(HvMV1) x⊕(y⊕z)∩(x⊕y)⊕z≠ ∅, (weakassociativity)
(HvMV2) x⊕y∩y⊕x≠ ∅, (weakcommutativity)
(Hv MV3) (x∗)∗=x,
33
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, University of Bojnord, Bojnord, Iran
[email protected], [email protected], [email protected]
(Hv MV4) (x∗ ⊕ y)∗ ⊕ y ∩ (y∗ ⊕ x)∗⊕ x ≠∅,
(HvMV5) 0∗∈x⊕0∗∩0∗⊕x,
(Hv MV6) 0∗ ∈ x ⊕ x∗ ∩ x∗ ⊕ x, (Hv MV7) x∈ x ⊕ 0∩ 0 ⊕ x,
(Hv MV8) 0∗ ∈ x∗ ⊕ y ∩ y ⊕ x∗ and 0∗ ∈ y∗ ⊕ x ∩ x ⊕ y∗ imply x = y.
Remark 2.2. On any Hv MV-algebra H, a binary relation ‘ ≼ ’ by
x ≼ y⇔ 0∗ ∈ x∗ ⊕ y ∩ y ⊕ x∗
is introduced which is reflexive and antisymmetric but not necessarily transitive.
HvMV-ideals
In this section, the ideal theory of HvMV-algebras is studied. The concepts of weak HvMV-
ideal and HvMV-ideal are introduced and some properties and fundamental results are
obtained.
Deftnition 3.1.Let I be a nonempty subset of H satisfying (𝐼0) x ≼ y and y ∈ I imply x ∈ I.
Then, I is called
1. an HvMV-ideal if x⊕y⊆I,for all x,y∈I,
2. a weak HvMV-idealif x⊕y≼ I, for all x,y ∈I.
Theorem 3.2. Every Hv MV-ideal is a weak Hv MV-ideal.
Theorem 3.4. A non empty subset I of H is a weak HvMV-ideal if and only if it satisfies (𝐼0)
and (x⊕y)∩I≠ ∅ , for all x,y∈I.
Theorem 3.7. If {Iα :α ∈ Λ} is a nonempty family of Hv MV-ideals of H, then ⋂ 𝐼𝛼∈Λ 𝛼 is a
HvMV-ideal.
Deftnition 3.8. Let A be a nonempty subset of H and {Iα : α ∈ Λ} be a family of Hv
MV-ideals of H containing A. Then ⋂ 𝐼𝛼∈Λ 𝛼 is called the Hv MV-ideal generated by A,
denoted by (A).
Remark 3.12. It seems that Theorem 3.7 does not hold for weak Hv MV- ideals, ingeneral. At
this point, we can’t find any example showing this(open problem). But, there is a situation in
which we know that the intersection of a family of weak Hv MV-ideals is again a weak Hv
MV-ideal (see Theorem 3.17). When the same result holds for weak Hv MV-ideals, the weak
Hv MV-ideal generated by a family A of weak Hv MV-ideals of H is denoted by (𝐴)𝑤 .
Let Hv MVI (WHv MVI ) denotes the set of all Hv MV-ideals (weak Hv MV- ideals) of H.
Then, HvMVI (WHv MVI ) together with the set inclusion, as a partial ordering, is a poset.
34
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, University of Bojnord, Bojnord, Iran
[email protected], [email protected], [email protected]
Theorem3.13.(HvMVI,⊆)is a complete lattice and if WHvMVI isclosed with respect to the
intersection, HvMVI is a complete sublattice of the complete lattice (WHv MVI,⊆).
Deftnition 3.14. An element a ∈ H is called a right scalar if for all x ∈ H, |x ⊕ a| = 1, i.e., the
set x ⊕ a is singleton. We denote the set of all right scalars of H by R(H).
Deftnition 3.16.A subset S of H is said to be ⊕-closed, if for all x, y ∈ S, x ⊕ y ⊆S.
Theorem 3.17. Assume that |x ⊕ y| <∞, for all x, y ∈ H, ≼ is transitive and monotone and
R(H) is ⊕-closed. If A is a nonempty subset of H contained in R(H), then
(𝐴)𝑤={x∈H: x ≼ (•••((𝑎1⊕𝑎2)⊕•••)⊕ 𝑎𝑛, for some n∈N, 𝑎1,..., 𝑎𝑛∈A}.
Particularly, if A = {a}, (𝑎)𝑤= {x ∈ H : x na, for some n ∈ N}.
Nodal HvMV-ideals
Deftnition 4.1. Let H be an HvMV-algebra. By a(weak)nodal HvMV-ideal of H we mean a
(weak) Hv MV-ideal of H which is comparable with each (weak) HvMV-ideal of H.
Example4.2.Consider the HvMV-algebra (H;⊕,∗,0), where ⊕ and ∗ are defined as in
Table4. Routine calculations show that the only HvMV-ideals of H are{0}, {0, a} and H. So,
every Hv MV-ideal of H is a nodal Hv MV-ideal.
⊕ 0 a b c 1
0 {0} {0,a} {0, b} {0,c} {0, a, b, c, 1}
a {0,a} {0,a} {0, a, b, c, 1} {0, a, b, c, 1} {0, a, b, c,1}
b {0, b} {0, a, b, c, 1} {0, a, b, c, 1} {0, a, b, c} {0, a, b, c,1}
c {0,c} {0, a, b, c, 1} {0, a, b, c} {0, a, b, c, 1} {0, a, b, c,1}
1 {0,a,b,c,1} {0,a,b,c,1} {0,a,b,c,1} {0,a,b,c,1} {0,a,b,c,1}
∗ 1 b a c 0
Table 4: Cayley table of Hv MV-algebra given in Example4.2
Proposition 4.5. Any nodal Hv MV-ideal is a nodal weak Hv MV-ideal.
Theorem 4.6. LetI be a(weak)HvMV-ideal of H. If for every x∈I and for every y∈H\I, x≼y,
then I is a nodal(weak)HvMV-ideal.
35
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, University of Bojnord, Bojnord, Iran
[email protected], [email protected], [email protected]
Theorem 4.7. In an HvMV-algebra with a totally ordered, any(weak)HvMV- ideal is a
nodal(weak)HvMV-ideal.
Theorem4.8. Assume that the conditions of Theorem3.17 holds for HvMV- algebra H. If x ∈
R(H) is a node, (𝑥)𝑤 is also a nodal weak Hv MV-ideal of H.
For weak Hv MV-ideal I of H and x ∈ H, the weak Hv MV-ideal of H Generated by I∪{x}
will denoted by I(x).
Theorem 4.10. Assume that the conditions of Theorem3.17 holds for HvMV- algebra H. If I
is a nodal weak Hv MV-ideal of H and x ∈ R(H) is a node, then I(x) is a nodal weak Hv MV-
ideal of H.
Theorem4.11. The intersection of any nonempty family of nodal HvMV- ideals is again a
nodalHvMV-ideal.
Considering Theorem3.13 we get
Corollary 4.12. Let N (H) be the set of all nodal Hv MV-ideals of H. Then (N (H), ⊆) is a
complete sublattice of (Hv MVI ,⊆).
Keyword(s): hyperstructures, algebras of logics.
2010 AMS Classification: 06F35, 03G25.
Reference(s):
1. Bakhshi M., Hv MV-algebras I, Quasigroups Related Systems,22, 9-18, 2014.
2. Bakhshi M. and Borzooei R.A.,Ordered polygroups, Ratio Mathematica, 24,31-40, 2013.
3. Chang C. C., Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88,
467-490, 1958.
4. Corsini P. and Leoreanu V., Applications of Hyperstructure Theory, Kluwer Academic
Publishers, Dordrecht, 2003.
5. Ghorbani Sh., Hassankhani A. and Eslami E., Hyper MV-algebras, Set-Valued
Mathematics and Applications, 1, 205-222, 2008.
6. Marty F., Sur une generalization de la notion de groups, 8th congress Math. Scandinaves,
Stockhholm, 45-49, 1934.
7. Mundici D., Interpretation of AFC∗-algebras in Łukasiewiczsententialcalsulus, J.
Func.Anal., 65, 15-63, 1986.
8. Vougiouklis T., A new class of hyperstructures, J. Combin. inform. Syst. Sci. 20, 229–235,
1995.
36
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Shahid Bahonar University, Kerman, Iran ,
2Centre for Systems and Information Technologies, University of Nova Gorica, Vipavska cesta 13, 5000 Nova Gorica, Slovenia,
3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic,
E-mails: [email protected]
Overview on the Height of a Hyperideal in Krasner
Hyperrings
Hashem Bordbar1, Irina Cristea
2, and Michal Novak
3
Similarly as in ring theory, the notion of height of a prime hyperideal of a hyperring
has recently been defined and studied [1], extending the concept of dimension of a hyperring,
in the context of commutative Krasner hyperrings. These are hyperstructures endowed with an
additive hyperoperation and a multiplicative operation, satisfying certain properties,
introduced by Krasner [7] as a tool in the approximation of valued fields. The height of a
proper prime hyperideal of a Krasner hyperring is defined as the maximum of the lengths of
the chains of distinct prime hyperideals contained in it, or it is ∞ if such a number does not
exist.
One of the most significant theorems in commutative algebra is called Krull’s height
theorem or Krull’s principal ideal theorem. Using the properties of prime hyperideals in
Noetherian Krasner hyperrings, we present an extension of this theorem [1]: If R is a
commutative Krasner hyperring and I is a proper principal hyperideal of R, then the height of
a minimal prime hyperideal of R over I is at most 1. Later on in [2], we extended this result,
proving that in a commutative Krasner hyperring R, the height of a minimal prime hyperideal
over a proper hyperideal of R generated by n elements is at most n. The converse of this
theorem is also true.
Our future goal is to extend these results to other classes of hyperrings, highlighting
their differences/similarities with the classical results for commutative rings.
Keywords: Krasner hyperring, prime/maximal hyperideal, Noetherian hyperring,
height of a prime hyperideal, dimension of a hyperring
2010 AMS Classification: 06F35, 03G25.
Reference(s):
1. H. Bordbar, I. Cristea, Height of prime hyperideals in Krasner hyperrings, Filomat
(accepted for pubblication in 2017).
2. H. Bordbar, I. Cristea, M. Novak, Height of hyperideals in Noetherian Krasner hyperrings,
U.P.B. Sci. Bull., Series A, 79(2017), no. 2, 31-42.
3. I. Cristea, S. Jancic-Rašovic, Composition hyperrings, An. St. Univ. Ovidius Constanta,
21(2013), no.2, 81-94.
37
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Shahid Bahonar University, Kerman, Iran ,
2Centre for Systems and Information Technologies, University of Nova Gorica, Vipavska cesta 13, 5000 Nova Gorica, Slovenia,
3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic,
E-mails: [email protected]
4. B. Davvaz, V. Leoreanu-Fotea, Hyperring theory and applications, International
Accademic Press, Palm Harbor, U.S.A., 2007.
5. D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New
York: Springer-Verlag, 1995.
6. I. Kaplansky, Commutative rings, University of Chicago Press, Chicago, 1974.
7. M. Krasner, Approximation des corps values complets de caracteristique p; p > 0, par ceux
de caracteristique zero, Colloque d’Algebre Superieure (Bruxelles, Decembre 1956), CBRM,
Bruxelles, 1957.
8. Ch.G. Massouros, On the theory of hyperrings and hyperfields, Algebra i Logika, 24(1985),
728-742.
9. J. D. Mittas, Hyperanneaux canoniques, Math. Balkanica, 2(1972), 165- 179.
10. A. Nakassis, Recent results in hyperring and hyperfield theory, Int. J. Math. Math. Sci.,
11(1988), 209-220.
11. S. Spartalis, A class of hyperrings, Riv. Mat. Pura Appl. 4 (1989), 55-64.
12. T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfields,
Algebraic hyperstructures and applications (Xanthi, 1990), 203–211, World Sci. Publ.,
Teaneck, NJ, 1991.
38
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Shahid Bahonar University, Kerman, Iran ,
2Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea
E-mail(s): [email protected]
Theory of Double-framed soft set theory on Hyper
BCK-algebra
Hashem Bordbar1 and Young Bae Jun
2
The notion of double-framed soft (strong) hyper BCK-ideal of hyper BCK-algebra is
introduced, and related properties are investigated. Characterization of double-framed soft
(strong) hyper BCK-ideal is considered, and relation between double-framed soft hyper BCK-
ideal and double-framed soft strong hyper BCK-ideal is discussed.
Keywords: Cubic intuitionistic set, cubic intuitionistic ideal, positive implicative
cubic intuitionistic ideal.
2010 AMS Classification: 06F35, 03G25, 06D72.
References:
1. Y. B. Jun and S. S. Ahn, Double-framed soft sets with applications in BCK/BCI-algebras,
J. Appl. Math. Volume 2012, Article ID 178159, 15 pages.
2. Y. B. Jun, G. Muhiuddin and A. M. Al-roqi, Ideal theory of BCK=BCI- algebras based on
double-framed soft sets, Appl. Math. Inf. Sci. 7(5), (2013), 1879–1887.
3. Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras,
Inform. Sci. 178, (2008), 2466–2475.
4. Y. B. Jun and X. L. Xin, Scalar elements and hyperatoms of Hyper BCK-algebras,
Scientiae Mathematicae 2(3), (1999), 303–309.
5. Y. B. Jun, X. L. Xin, E. H. Roh and M. M. Zadehi, Strong hyperBCK- ideals of
hyperBCK-algerbas, Math. Japon. 51(3), (2000), 493–498.
39
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Democritus University of Thrace, School of Education, Alexandroupolis, Greece
2 Democritus University of Thrace, School of Education, Alexandroupolis, Greece
E-mail(s):[email protected], [email protected]
On P-hopes and P-Hv-structures on the plane
Achilles Dramalidis1, Ioanna Iliou
2
In this paper we deal with P-hyperstructures which are defined in Hv-
groups. Using a weak commutative hyperoperation on the plane and a
specific subset of this plane we construct various P-Hv-structures. In addition,
we study the existence of units and inverses of these constructions,
connecting them with Join Spaces, as well.
Keyword(s): Hyperstructures, Hv-structures, hopes, P-hyperstructures.
2010 AMS Classification: 20N20
References:
1. Antampoufis N., Vougiouklis T., Hyperoperations greater than the Complex Number
operations, Journal of Basic Science, 3, 11-17, 2006.
2. Corsini P., Leoreanu V., Application of Hyperstructure Theory, Klower Ac.
Publ., 2003.
3. Davvaz B., Santilli R.M., Vougiouklis T., Studies of multivalued hyper-
structures for the characterization of matter-antimatter systems and their
extension, Algebras, Groups and Geometries 28, 105–116, 2011.
4. Dramalidis A., Some geometrical P-HV-structures, New Frontiers in
Hyperstructures, 93-102, 1996.
5. Dramalidis A., Dual Hv-rings, Rivista di Matematica Pura ed Applicata, 17,
55-62, 1996.
6. Dramalidis A., Vougiouklis T., Fuzzy Hv-substructures in a two dimensional
Euclidean vector space, Iranian J. Fuzzy Systems, 6, 1-9, 2009.
7. Iranmanesh A., Iradmusa M.N., Hv-structures associated with generalized P-
hyperoperations, Bul. of the Iranian Math. Soc., 24, 33–45, 1998.
8. Vougiouklis T., Generalization of P-hypergroups, Rendiconti Circolo
Matematico di Palermo, 36, 114-121, 1987.
9. Vougiouklis T., The fundamental relation in hyperrings. The general
hyperfield, World Scientific, 203-211, 1991.
10. Vougiouklis T., Hyperstructures and their Representations, Monographs
Math., Hadronic Press, 1994.
11. Vougiouklis T., Representations of hypergroups by generalized permutations,
Algebra Universalis, 29, 172-183, 1992.
12. Vougiouklis T., Dramalidis A., Hv-modulus with external P-hyperoperations,
Proc. of the 5th AHA, Iasi, Romania, 191-197, 1993.
40
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Kocaeli University, Department of Mathematics, Kocaeli, Turkey
2 Kocaeli University, Department of Mathematics, Kocaeli, Turkey
3 Kocaeli University, Department of Mathematics, Kocaeli, Turkey
E-mails:[email protected], [email protected], [email protected]
On Neutrosophic Linear Spaces
Banu Pazar Varol 1, Vildan Çetkin
2, Halis Aygün
3
Smarandache introduced the neutrosophy which is a branch of philosophy. Then Wang
et.al. defined single valued neutrosophic sets. Neutrosophic set is a part of neutrosophy which
studies the origin, nature and scope of neutralities. In neutrosophic set, truth-membership,
indeterminacy membership and false-membership functional values are independent. Single
valued neutrosophic set is applied to algebraic and topological structures. In this paper, we
introduce neutrosophic linear space over the neutrosophic field and consider its main
properties.
Keywords: Neutrosophic set, single valued neutrosophic set, linear space
2010 AMS Classification: 08A72, 06D72
References:
1. Nanda S., Fuzzy fileds and fuzzy linear spaces, Fuzzy sets and Systems, 19, 89-94, 1986.
2. Smarandache F., A unifying field in logics. Neutrosophy/ Neutrosophic Probability, Set
and Logic, Rehoboth: American Research Press (1998) http://fs.gallup.unm.edu/eBook-
neutrosophics6.pdf (last edition online).
3. Wang H. et al., Single valued neutrosophic sets, Proc. of 10th Int. Conf. on Fuzzy Theory
and Technology, Salt Lake City, Utah, July, 21-26, 2005.
41
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Recep Tayyip Erdoğan University, Department of Mathematics, Rize, Turkey
E-mail:[email protected]
The Relationships between the Orders Induced by Implications and Uninorms
M. Nesibe Kesicioğlu1
In this paper, an order by means of implications on a bounded lattice possessing some
special properties is defined and some of its properties are discussed. By giving an order
based on uninorms on a bounded lattice, the relationships between such generated orders are
investigated.
Keywords: Implications, partial order, bounded lattice, law of importation
2010 AMS Classification: 03E72, 03B52
References:
1. Baczynski M., Jayaram B., Fuzzy implications, Studies in Fuzziness and Soft Computing,
vol. 231, Springer, Berlin, Heidelberg, 2008.
2. Birkhoff G., Lattice Theory, 3 rd edition, Providence, 1967.
3. Jayaram B., A new ordering based on fuzzy implications, Proceedings of ISAS 2016,
Luxembourg, Luxembourg, 49-50, 2016.
4. Karaçal F., Kesicioğlu M.N., A T-partial order obtained from t-norms, Kybernetika, 47,
300-314, 2011.
5. Kesicioğlu M.N., Mesiar R., Ordering based on implications, Information Sciences, 276,
377-386, 2014.
6. Mas M., Monserrat M., Torrens J., A characterization of (U,N), RU, QL and D-
implications derived from uninorms satisfying the law of importation, Fuzzy Sets and
Systems, 161, 1369-1387, 2010.
42
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Recep Tayyip Erdoğan University, Department of Mathematics, Rize, Turkey 2Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey 3Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey
E-mails: [email protected], [email protected], [email protected]
A Survey on Order-equivalent Uninorms
M. Nesibe Kesicioğlu1, Ümit Ertugrul
2, Funda Karaçal
3
In this paper, an equivalence on the class of uninorms on a bounded lattice 𝐿 based on
the equality of the orders is discussed. Some relationships between the orders induced by t-
norms and their N-dual t-conorms are determined. Also, defining the set of all incomparable
elements w.r.t. the order induced by uninorms, some relationships with the sets of all
incomparable elements w.r.t. the orders induced by the corresponding underlying t-norm and
t-conorm are presented.
Keywords: Uninorm, bounded lattice, partial order, equivalence of uninorms
2010 AMS Classification: 03E72, 03B52
References:
1. Birkhoff G., Lattice Theory, 3 rd edition, Providence, 1967.
2. Ertuğrul Ü, Kesicioğlu M.N., Karaçal F., Ordering based on uninorms, Information
Sciences, 330, 315-327, 2016.
3. Grabisch M., Marichal J.-L., Mesiar R., Pap E., Aggregation Functions, Cambridge
University Press, 2009.
4. Karaçal F., Kesicioğlu M.N., A T-partial order obtained from t-norms, Kybernetika, 47,
300-314, 2011.
5. Kesicioğlu M.N., Karaçal F., Mesiar R., Order-equivalent triangular norms, Fuzzy Sets and
Systems, 268, 59-71, 2015.
6. Kesicioğlu M.N., Mesiar R., Ordering based on implications, Information Sciences, 276,
377-386, 2014.
7. Klement E.P., Mesiar R., Pap E., Triangular Norms, Kluwer Academic Publishers,
Dordrecht, 2000.
43
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 State University of Moldova 2State University from Tiraspol
E-mails: [email protected], [email protected]
THE REFLECTOR FUNCTOR AND THE LATTICE 𝕃(𝓡)
Cerbu Olga1 and Dumitru Botnaru
1
In the category 𝒞2𝒱 of locally convex topological vector spaces [RR] we examine a
class of factorization structures for which the reflector functor transforms the class of
projections or the class of injections, or both classes into themselves. Such functors are
usually studied (see [K], [G], [B], [BC], [C]) and appear when studying the semireflexive
subcategories [CB].
Let 𝛱 be the subcategory of the complete spaces with the weak topology and
𝜋: 𝒞2𝒱 → 𝛱 - the reflector functor. The subcategory 𝛱 is the minimal element in the lattice ℝ.
Let ℛ ∈ ℝ. For every object 𝑋 of the category 𝒞2𝒱 let be 𝑟𝑋: 𝑋 → 𝑟𝑋 and 𝜋𝑋: 𝑋 → 𝜋𝑋 the ℛ
and 𝛱-repliques. Since 𝛱 ⊂ ℛ, we have
𝜋𝑋 = 𝑣𝑋𝑟𝑋
for a morphism 𝑣𝑋 . We denote 𝒰 = 𝒰(ℛ) = {𝑟𝑋 ∣ 𝑋 ∈∣ 𝒞2𝒱 ∣}, 𝒱 = 𝒱(ℛ) = {𝑣𝑋 ∣
𝑋 ∈∣ 𝒞2𝒱 ∣}. We have the following factorization structures:
, , , , , ., ,P P P P └ └ └
For ℛ ∈ ℝ we denote by 𝕃(ℛ) the class of factorization structures (ℰ,ℳ), for which
𝒫ʹ(ℛ) ⊂ ℰ ⊂ 𝒫ʺ(ℛ) and 𝕃𝑢(ℛ) = {(ℰ,ℳ) ∈ 𝕃(ℛ) ∣ ℳ ⊂ℳ𝑢} (see [B]), where ℳ𝑢 is a
class of the universal monomorphisms (see [B]).
.,p uu u uP P
└
Definition 1. Let 𝑟: 𝒞 → ℛ be a covariant functor, and (𝒫, ℐ) - a factorization structure (the
left or right factorization structure). We say that this functor 𝑟 is:
1. P-functor, if r(P) P.
2. I-functor, if r(I) I.
3. (P; I)-functor, if r(P) P and r(I) I.
Proposition 2. 1. 𝕃𝑢(ℛ) is a complete lattice with the minimal element (𝒫ʹ𝑢, ℐʹ𝑢) and the
maximal element (𝒫ʺ, ℐʺ).
2. 𝕃𝑢(ℛ) is the class of the factorization structures (ℰ,ℳ), for which ℐʹ𝑢 ⊂ ℳ ⊂ ℐʺ.
Lemma 3. Let 𝑚:𝑋 → 𝑌 be an universal monomorphism. Then 𝜋(𝑚) is a sectional
morphism.
Theorem 4. 1. Let (ℰ,ℳ) ∈ 𝕃𝑢(ℛ). Then 𝑟: 𝒞2𝒱 → ℛ is a (ℰ,ℳ)-functor: 𝑟(ℰ) ⊂ ℰ and
𝑟(ℳ) ⊂ ℳ.
2. 𝑓 ∈ 𝒫ʺ(ℛ) ⇔ 𝑟(𝑓) ∈ 𝒫ʺ(ℛ).
44
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 State University of Moldova 2State University from Tiraspol
E-mails: [email protected], [email protected]
Keywords: Reflector functor, lattice
2010 AMS Classification: 55P65
References:
1. [B] Botnaru D., Structures bicatégorielles complémentaires. ROMAI J., v.5, Nr.2, 2009,
p.5-27.
2. [BC] Botnaru D., Cerbu O., Functor of Special Type, Proceedings of the VIII International
Workshop, Lie Theory and its Applications in Physics, Bulgaria Academy of Sciences,
Institute for Nuclear Reasearch and Nuclear Energy, Varna, Bulgaria, 15-21 June 2009, p.
299-311
3. [CB] Cerbu O., Botnaru D., Some properties of semireflexivity, Noncommutative structures
in mathematics and physics, 22-26 July 2008, Brussels, p.71-84.
4. [C] Cerbu O., The Lattice of Semireflexive Subcategories, ROMAI Journal, CAIM,
Universitatea Al. Ioan Cuza, Septembrie 2010, Iaşi
5. [G] Grothendieck A., Topological vector spaces, Gordon and Breach, 1973.
6. [K] Kennison J. F., Reflective functors in general topology, TAMS, 1965, v.118, p.309-
315.
7. [RR] Robertson A. P., Robertson W. J., Topological vector spaces, Cambridge, England,
1964.
45
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails:,[email protected], [email protected], [email protected], [email protected]
2-Absorbing -Primary Fuzzy Ideals of Commutative Rings
Deniz Sönmez1, Gürsel Yeşilot
1, Serkan Onar
1 and Bayram Ali Ersoy
1
In this work, we define 2-Absorbing -primary fuzzy ideals which is the
generalizations of 2-absorbing fuzzy ideal and 2-absorbing primary fuzzy ideals. Furthermore,
we give some fundamental results concerning these notions.
Keywords: 2-Absorbing -Primary fuzzy ideals, 2-Absorbing primary fuzzy ideals.
2010 AMS Classification: 03E72
References:
1.Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.
2.A. Badawi , On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3,
417-429.
3. A. Badawi, U. Tekir, E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Austral.
Math. Soc. 51 (2014), no. 4, 1163-1173.
4. F. Callialp, E. Yetkin and U. Tekir, On 2-absorbing primary and weakly 2-absorbing primary
elements in multiplicative lattices, Italian Journal of Pure and Applied Mathematics, 34 (2015), 263-
276 .
5. Badawi A. and Darani A.Y, On weakly 2-absorbing ideals of commutative rings. Houston J. Math.
39, 441-452, 2013.
6. Malik D.S. and Mordeson J.N., Fuzzy Commutative Algebra, World Scienti_c Publishing, 1998.
7. V.N. Dixit, R. Kumar and N. Ajmal. Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy Sets and
Systems 44 (1991), 127-138.
8.W.J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982), 133-139.
9.T.K. Mukherjee and M.K. Sen, Prime fuzzy ideals in rings, Fuzzy Sets and Systems 32 (1989), 337-
341.
10. T.K. Mukherjee and M.K. Sen, Primary fuzzy ideals and radical of fuzzy ideals, Fuzzy Sets and
Systems 56 (1993), 97-101.
11.L.I. Sidky , S.A. Khatab, Nil radical of fuzzy ideal, Fuzzy Sets and Systems 47 (1992), 117-120.
46
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, Shahid Beheshti University, Tehran, Iran
2Department of Mathematics, Alzahra University, Tabnak Street, Vanak Square,
E-mails: [email protected] , [email protected]
Relation Between Hyper EQ-algebras and Some Other Hyper Structures
Rajab Ali Borzooei1 and Batol Ganji Saffar
2
In this study by considering the notion of hyper EQ-algebra, as a generalization
of EQ-algebra (algebra of truth values for a higher-order fuzzy logic), we define some types
of filters in this structuer and investigated some related results. Then we find the relation
between hyper EQ-algebras and hyper BCK-algebras, hyper MV -algebras and (weak) hyper
residuated lattices.
Keywords: Hyper EQ-algebra, hyper BCK-algebra, hyper MV -algebra, (weak) hyper
residuated lattice
2010 AMS Classification: 20N20
References:
47
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1,2 Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey 3 Yazd University, Department of Mathematics, Yazd, Iran
E-mail(s): [email protected] , [email protected] , [email protected]
Fuzzy hyperideals in ordered semihyperrings
Osman KAZANCI1, Şerife YILMAZ
2, Bijan DAVVAZ
3
In this study, we introduce the concept of fuzzy hyperideals of ordered
semihyperrings, which is a generalization of the concept of fuzzy hyperideals of
semihyperrings to ordered semihyperring theory. We investigate its related properties. We
show that every fuzzy quasi-hyperideal is a fuzzy bi-hyperideal and in a regular ordered
semihyperring, fuzzy quasi-hyperideal and fuzzy bi-hyperideal coincide.
Keywords: Semihyperring: ordered semihyperring: fuzzy hyperideal.
2010 AMS Classification: 03E72; 97H50.
Reference(s):
1. Ameri R., Hedayati H., On k-hyperideals of semihyperrings, J. Discrete Math. Sci.
Cryptogr., 10(1), 41-54, 2007.
2. Corsini P., Leoreanu-Fotea V., Applications of Hyperstructure Theory, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2003.
3. Davvaz B., Cristea I., Fuzzy Algebraic Hyperstructures, An Introduction, Springer, 2015.
4. Davvaz B., Leoreanu-Fotea V., Fuzzy ordered Krasner hyperrings, Journal of Intelligent
and Fuzzy Systems, 29, 2015.
5. Heidari D., Davvaz B., On ordered hyperstructures, Politech. Univ. Bucharest Sci. Bull.
Ser. A. Appl. Math. Phys., 73(2), 85-96, 2011.
6. Kazancı O., Yamak S., Generalized fuzzy bi-ideals of semigroup, Soft Computing, 12,
1119-1124, 2009.
7. Kehayopulu N., Tsingelis M., Fuzzy Right, Left, Quasi-Ideals, Bi-Ideals in Ordered
Semigroups, Lobachevskii Journal of Mathematics, 30, 17-22, 2009.
8. Rosenfeld A., Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.
9. Vougiouklis T., Hyperstructures and their Representations, Hadronic Press, Florida, USA,
1994.
10. Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.
48
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1,2 Karadeniz Technical University, Department of Mathematics, Trabzon, Turkey
E-mail(s): [email protected] , [email protected]
Fuzzy interior hyperideals in ordered semihyperrings
Şerife YILMAZ 1, Osman KAZANCI
2
In this study, we introduce the concept of fuzzy interior hyperideals in ordered
semihyperrings, which are a new sort of fuzzy hyperideals of semihyperrings. We investigate
some of their related properties. We give a characterization of fuzzy interior hyperideals in
terms of their level subsets. We show that every fuzzy hyperideal is a fuzzy interior
hyperideal. We introduce the concept of intra-regular ordered semihyperrings and show that
fuzzy hyperideals and fuzzy interior hyperideals coincide in an intra-regular ordered
semihyperring. Finally, we introduce the concept of fuzzy simple ordered semihyperrings and
prove some results.
Keywords: Ordered semihyperring: interior hyperideal: fuzzy interior hyperideal.
2010 AMS Classification: 03E72; 97H50.
Reference(s):
1. Ameri R., Hedayati H., On k-hyperideals of semihyperrings, J. Discrete Math. Sci.
Cryptogr., 10(1), 41-54, 2007.
2. Corsini P., Leoreanu-Fotea V., Applications of Hyperstructure Theory, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2003.
3. Davvaz B., Cristea I., Fuzzy Algebraic Hyperstructures, An Introduction, Springer, 2015.
4. Hedayati G., t-implication-based fuzzy interior hyperideals of semihypergroups, Journal of
Discrete Mathematical Sciences and Cryptography, 13(2), 123-140, 2010.
5. Kazancı O., Yamak S., Generalized fuzzy bi-ideals of semigroup, Soft Computing, 12,
1119-1124, 2009.
6. Kehayopulu N., Tsingelis M., Fuzzy Right, Left, Quasi-Ideals, Bi-Ideals in Ordered
Semigroups, Lobachevskii Journal of Mathematics, 30, 17-22, 2009.
7. Rosenfeld A., Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.
8. Tang J., Davvaz B., Luo Y., A study on fuzzy interior hyperideals in ordered
semihypergroups, Italian Journal of Pure and Applied Mathematics, 36, 125-146, 2016.
9. Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.
49
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Iran
E-mails: [email protected], [email protected]
HYPERHILBERT SPACES
SAEID GHOLAMPOOR1 and ALI TAGHAVI
1
In this study we introduce the concept of hyper Hilbert spaces and prove some result
such as, orthogonal projection and Riesz theorem about them.
Keywords: Hyperhilbert space, hilbert space, hyper space
2010 AMS Classification: 20N20
References:
1. P. Corsini, Prolegomena of hypergroup theory, Aviani editore, (1993).
2. P. Corsini and V. Leoreanu, Applications of Hyperstructure theory, Kluwer Academic Publishers,
Advances in Mathematics (Dordrecht), (2003).
3. F. Marty, Sur nue generalizeation de la notion de group, 8th congress of the Scandinavic
Mathematics, Stockholm, (1934), 45{49.
4. S. Roy and T. K. Samanta, Innerproduct hyperspaces, Accepted in Italian J. of Pure and Appl. Math.
5. A. Taghavi and R. Hosseinzadeh, A note on dimension of weak hypervector spaces, Italian J. of
Pure and Appl. Math, To appear.
6. A. Taghavi and R. Hosseinzadeh, Hahn-Banach Theorem for functionals on hypervector spaces,
The Journal of Mathematics and Computer Science, Vol .2 No.4 (2011) 682-690.
7. A. Taghavi and R. Hosseinzadeh, Operators on normed hypervector spaces, Southeast Asian
Bulletin of Mathematics, (2011) 35: 367-372.
8. A. Taghavi and R. Hosseinzadeh, Operators on weak hypervector spaces, Ratio Mathematica, 22
(2012) 37-43.
9. A. Taghavi and R. Hosseinzadeh and H. Rohi, Hyperinner product spacess,
10. A. Taghavi and T. Vougiouklis and R. Hosseinzadeh, A note on Operators on Normed Finite
Dimensional Weak Hypervector Spaces, Scientic bulletin, Series A, Vol. 74, Iss. 4 (2012) 103-108.
11. M.Scafati-Tallini, Characterization of remarkable Hypervector space, Proc. 8th congress on
"Algebraic Hyperstructures and Aplications", Samotraki, Greece, (2002), Spanidis Press, Xanthi,
(2003), 231-237.
12. M.Scafati-Tallini, Weak Hypervector space and norms in such spaces, Algebraic Hyperstructures
and Applications Hadronic Press. (1994), 199-206.
13. T. Vougiouklis, The fundamental relation in hyperrings. The general hyper_eld. Algebraic
hyperstructures and applications (Xanthi, 1990), World Sci. Publishing, Teaneck, NJ, (1991),
203{211.
14. T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, (1994).
15. M. M. Zahedi, A review on hyper k-algebras, Iranian Journal of Mathematical Sciences and
Informatics, Vol. 1, No. 1 (2006). 55-112.
50
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Namık Kemal Univesity, Department of Mathematics, Tekirdağ, Turkey.
2 Karadeniz Technical Univesity, Department of Mathematics, Trabzon, Turkey.
E-mail(s): [email protected], [email protected], [email protected].
The Lattice Structure of Subhypergroups of a Hypergroup
Dilek BAYRAK1, Sultan YAMAK
2, Şerife YILMAZ
2
In mathematics, determination of algebraic structures is very important. Many
methods have been applied in determining these structures until today. One of them is
investigating the lattice structure of substructures of algebraic structures (such as submonoids
of a monoid, subgroups of a group, ideals of a ring, submodules of a module, subspaces of a
vector space, etc.) according to the inclusion relation. As a generalization of algebraic
structures, hyper structure was defined in 1934 by F. Marty. Since then this theory has been
developed by many mathematicians. In the last fifteen years, various applications of algebraic
structures (in geometry, binary relations, lattices, fuzzy sets, rough sets, automata,
cryptography, codes, median algebra, relational algebra, artificial intelligence probability)
have been obtained.
In this study, we investigate the properties of closed, invertible, ultraclosed and
conjugable subhypergroups classes. We study when the hypergroups satisfy the property that
the hyperproduct of subhypergroups becomes an operation on the set of subhypergroups. It is
investigated in which cases, the poset of the subhypergroups of a hypergroup is a lattice. It is
examined when this lattice is modular or distributive. Thus some information about a
hypergroup may be obtained by investigating the lattice of its subhypergroups.
Keyword(s): subhypergroups, lattice.
2010 AMS Classification: 20N20, 06B99.
Reference(s):
1. Birhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Publ., 1967.
2. Corsini P. and Leoreanu V., Application of Hyperstructure Theory, Kluwer Academic
Publishers, 2003.
3. Davvaz B., Leoreanu V., Hyperring theory and applications, International Academic Press,
2007.
4. Marty, F., Sur ungeneralisation de la notion degroup, 8th Congress of Scandinavian
Mathematicians, 45-49, 1934.
5. Massouros, C.G., Some properties of certain subhypergroups, Ratio Mathematica, 25, 67-
76, 2013.
6. Schmidt R., Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics 14, de
Gruyter, Berlin, 1994.
7. Tarnauceanu, M., On the poset of subhypergroups of a hypergroup, Int. J. Open Problems
Comp. Math. 3(2), 115-122, 2010.
51
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
Emails: [email protected], [email protected], [email protected], [email protected]
Intuitionistic Fuzzy Weakly Prime Ideals
Tuğba Arkan1, Serkan Onar
1, Deniz Sönmez
1, Bayram Ali Ersoy
1
In this study, the fundamental definitions and theorems regarding intuitionistic fuzzy
sets and intuitionistic fuzzy ideals of commutative ring with identity 𝑅 have been given as
preliminaries. After the preliminaries, we introduce the notions of intuitionistic fuzzy weakly
prime ideals, intuitionistic fuzzy partial weakly prime ideals, intuitionistic fuzzy weakly
semiprime ideals of 𝑅. Let 𝑃 = ⟨𝜇𝑃, 𝜐𝑃 ⟩ be a nonconstant intuitionistic fuzzy ideal of 𝑅. If
(0,1) ≠ 𝐴 ∙ 𝐵 ⊆ P implies 𝐴 ⊆ 𝑃 or 𝐴 ⊆ 𝑃 where 𝐴 = ⟨𝜇𝐴, 𝜐𝐴 ⟩ , 𝐵 = ⟨𝜇𝐵, 𝜐𝐵 ⟩ intuitionistic
fuzzy ideals of 𝑅, then 𝑃 is called intuitionistic fuzzy weakly prime ideal of 𝑅. If 𝑃(𝑥𝑦) =
𝑃(𝑥) or 𝑃(𝑥𝑦) = 𝑃(𝑦) for 𝑥𝑦 ≠ 0, then 𝑃 is called intuitionistic fuzzy partial weakly prime
ideal of 𝑅. A nonconstant intuitionistic fuzzy ideal 𝑃 is called intuitionistic fuzzy weakly
semiprime ideal of 𝑅 if (0,1) ≠ 𝐵2 ⊆ 𝑃 implies 𝐵 ⊆ 𝑃 where 𝐵 is an intuitionistic fuzzy
ideal of 𝑅. Also, we give some relations between intuitionistic fuzzy weakly prime ideals and
weakly prime ideals of 𝑅.
Keywords: Intuitionistic fuzzy prime ideals, intuitionistic fuzzy weakly prime ideals,
intuitionistic fuzzy partial weakly prime ideals, intuitionistic fuzzy weakly semiprime ideals.
2010 AMS Classification: 03F55, 03E72, 08A72.
References:
1. Atanassov K. , Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87–96, 1986.
2. Hur, K., Jang, S. Y. & Kang, H. W., Intuitionistic fuzzy ideal of a ring, J. Korea Soc. Math.
Educ. Ser. B: Pur Appl. Math., Vol 12, 2005.
3. P. A. Ejegwa, A. J. Akubo, O. M. Joshua, Intuitionistic Fuzzy Set and Its Application in
Career Determination via Normalized Euclidean Distance Method, European Scientific
Journal edition vol.10, No.15, 2014.
4. D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math., 29, 4, 831–840,
2003.
5. J.N Mordeson, D.S. Malik, Fuzzy Commutative Algebra, World Scientific Publishing.
R.Majoob, On Weakly Prime L-Ideals, Italian Journal of Pure and Applied Mathematics-N.
36-2016(465-472).
6. Zadeh L. A., Fuzzy sets, Inform. and Control, 8, 338–353, 1965.
7. Zahedi M. M., A Note On L-Fuzzy Primary and Semiprime Ideals, Fuzzy Sets and Systems
51 (2): 243-247, 1992.
52
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, Statistic and Computer Sciences, University of Tehran 2Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
E-mail(s): [email protected], [email protected]
Some Results on Tensor Product of Krasner Hypervector Spaces
R. Ameri1, K. Ghadimi
2
We introduce and study tensor product of Krasner hypervector spaces (Krasner hyperspac
es). Here we introduce the (resp. multivalued) middle linear maps of Krasner hyperspaces and
construct the categories of linear maps and multivalued linear maps of Krasner hyperspaces. It
is shown the tensor product of two Krasner hypespaces, as an initial object in this
category, exists. Also, notion of a quasi-free object in category of Krasner hyperspaces are
introduced and it is proved that in this category a quasi-free object up to maximum is unique.
Keyword(s): Krasner hypervector space, Multivalued middle linear map, Quasi-free, Tensor
product
2010 AMS Classification: 20N20
1 Introduction
The theory of algebraic hyperstructures is a well-established branch of classical algebraic
theory. Hyperstructure theory was first proposed in 1934 by Marty, who defined hypergroups
and began to investigate their properties with applications to groups, rational fractions and
algebraic functions [10]. It was later observed that the theory of hyperstructures has many
applications in both pure and applied sciences; for example, semihypergroups are the simplest
algebraic hyperstructures that possess the properties of closure and associativity. The theory
of hyperstructures has been widely reviewed ([6], [7], [8], [9] and [12]) (for more see [1, 2, 3,
4, 5]).
In [11] M. Motameni et. al. studied hypermatrix. R. Ameri in [1] introduced and studied
categories of hypermodules. Let 𝑉 and 𝑊 be two Krasner hyperspaces over the hyperfiled 𝐾.
The purpose of this paper is the study of tensor product of Krasner hyperspaces. We introduce
the category of multivalued linear maps of Krasner hyperspaces and then construct the tensor
product of 𝑉 and 𝑊 as initial object in this category.
2 Preliminaries and main results
Let 𝐻 be a nonempty set. A map ∙ ∶ 𝐻 × 𝐻 → 𝑃∗(𝐻) is called hyperoperation or join
operation, where 𝑃∗(𝐻) is the set of all nonempty subsets of 𝐻 . The join operation is
extended to nonempty subsets of 𝐻 in natural way, so that 𝐴 ∙ 𝐵 is given by
53
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, Statistic and Computer Sciences, University of Tehran 2Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
E-mail(s): [email protected], [email protected]
𝐴 ∙ 𝐵 = ⋃{𝑎 ∙ 𝑏 | 𝑎 ∈ 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵}.
the notations 𝑎 ∙ 𝐴 and 𝐴 ∙ 𝑎 are used for {𝑎} ∙ 𝐴 and 𝐴 ∙ {𝑎} respectively. Generally, the
singleton {𝑎} is identified by its element 𝑎.
Definition 1. [7] A semihypergroup (𝐻, +) is called a canonical hypergroup if the following
conditions are satisfied:
(i) 𝑥 + 𝑦 = 𝑦 + 𝑥 for all 𝑥, 𝑦 ∈ 𝑅;
(ii) There exists 0 ∈ 𝑅 (unique) such that for every 𝑥 ∈ 𝑅, 𝑥 ∈ 0 + 𝑥 = 𝑥;
(iii) For every 𝑥 ∈ 𝑅, there exists a unique element, say 𝑥′ such that 0 ∈ 𝑥 + 𝑥′
(we denote 𝑥′ = −𝑥);
(iv) For every 𝑥, 𝑦, 𝑧 ∈ 𝑅, 𝑧 ∈ 𝑥 + 𝑦 ⟺ 𝑥 ∈ 𝑧 − 𝑦 ⟺ 𝑦 ∈ 𝑧 − 𝑥;
from the definition it can be easily verified that −(−𝑥) = 𝑥 and −(𝑥 + 𝑦) = −𝑥 − 𝑦.
Definition 2. [7] Let (𝐾, +, ∗) be a hyperfield and (𝑉, +) be a canonical hypergroup. We
define a Krasner hyperspace over K to be the quadruplet (V, +, ∙, K) where ∙ is a single-
valued operation
∙ ∶ K × V → V
such that for all a ∈ K and x ∈ V we have a ∙ x ∈ V , and for all a, b ∈ K and x, y ∈ V the
following conditions are satisfied:
(H1) a ∙ (x + y) = a ∙ x + a ∙ y;
(H2) (a + b) ∙ x = a ∙ x + b ∙ x;
(H3) a ∙ (b ∙ x) = (a ∗ b) ∙ x;
(H4) 0 ∙ x = 0;
(H5) 1 ∙ x = x.
Remark 1. For simplicify, we say 𝑉 is a 𝐾𝑟-hyperspace.
54
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, Statistic and Computer Sciences, University of Tehran 2Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
E-mail(s): [email protected], [email protected]
Definition 4. Let (𝐹, ·) be an object in the category 𝐾𝑟 − H𝑣𝑒𝑐𝑡 and 𝑖 ∶ 𝑋 ↪ 𝐹 be an
inclusion map of sets. We say that 𝐹 is quasi-free on the subset X provided that:
(i) 𝐹 = ⟨𝑋⟩;
(ii) For any object 𝑉 in 𝐾𝑟 − 𝐻𝑣𝑒𝑐𝑡 and any multivalued map 𝜆 ∶ 𝑋 → 𝑃∗(𝑉 ), there is a
maximum 𝑠𝑚𝑣, �̅� ∶ 𝐹 → 𝑃∗(𝑉 ) such that for all 𝑥 ∈ 𝑋, we have �̅�𝑖(𝑥) = 𝜆(𝑥).
Definition 5. Let 𝑉 and 𝑊 be two 𝐾𝑟-hyperspaces over a hyperfield 𝐾 . Let F be the free
abelian group on the set 𝑉 × 𝑊. Let 𝐻 be the subgroup of 𝐹 generated by all elements of the
following forms (for all 𝑣, 𝑣′ ∈ 𝑉, 𝑤, 𝑤′ ∈ 𝑊, and 𝑎 ∈ 𝐾):
(i) (𝑣 + 𝑣′, 𝑤) − (𝑣, 𝑤) − (𝑣′, 𝑤), where (𝑣 + 𝑣′, 𝑤) =∪𝑡∈𝑣+𝑣′ (𝑡, 𝑤)
(ii) (𝑣, 𝑤 + 𝑤′) − (𝑣, 𝑤) − (𝑣, 𝑤′);
(iii) (𝑎 · 𝑣, 𝑤) − (𝑣, 𝑎 · 𝑤).
The quotient group 𝐹/𝐻 is called a tensor product of 𝑉 and 𝑊; it is denoted 𝑉 ⊗𝐾 𝑊. The
coset (𝑣, 𝑤) + 𝐾 of the element (𝑣, 𝑤) in 𝐹 is denoted 𝑣 ⊗ 𝑤 ; the coset of (0, 0) is
denoted 0.
Theorem 1. Let 𝐹 be a 𝐾𝑟-hyperspace over a hyperfield 𝐾 and 𝑋 be a basis for 𝐹. Then
(i) If 𝑗 ∶ 𝑋 ↪ 𝐹 is a inclusion map, then for all 𝐾𝑟-hyperspace 𝑉 and map f ∶ 𝑋 → 𝑃∗(𝑉 ),
there is a maximum 𝑠𝑚𝑣, 𝜑 ∶ 𝐹 → 𝑃∗(𝑉 ) such that the diagram
is commutative.
(ii) For all 𝐾𝑟-hyperspace 𝑉 and f ∶ 𝑋 → 𝑃∗(𝑉 ) induced maximum 𝑠𝑚𝑣, 𝜑 ∶ 𝐹 → 𝑃∗(𝑉 ),
means there is a maximum 𝑠𝑚𝑣, 𝜑 ∶ 𝐹 → 𝑃∗(𝑉 ) such that 𝜑|𝑋 = 𝑓.
55
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, Statistic and Computer Sciences, University of Tehran 2Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran
E-mail(s): [email protected], [email protected]
Given hyperspaces 𝑉 and 𝑊 over a hyperfield 𝐾, it is easy to verify that the map 𝑖 ∶ 𝑉 ×
𝑊 → 𝑉 ⊗𝐾 𝑊 given by (𝑣, 𝑤) ⟼ 𝑣 ⊗ 𝑤 is a middle linear map. The map i is called the
canonical middle linear map. Its importance is seen in
Theorem 2. Let 𝑉 and 𝑊 be 𝐾𝑟-hyperspaces over a hyperfield 𝐾, and let 𝑍 be an abelian
group. If 𝑔 ∶ 𝑉 × 𝑊 → 𝑍 is a middle linear map, then there exists unique group
homomorphism �̅� ∶ 𝑉 ⊗𝐾 𝑊 → 𝑍 such that �̅�𝑖 = 𝑔 , where 𝑖 ∶ 𝑉 × 𝑊 → 𝑉 ⊗𝐾 𝑊 is the
canonical middle linear map. 𝑉 ⊗𝐾 𝑊 is uniquely determined up to isomorphism by this
property. In other words 𝑖 ∶ 𝑉 × 𝑊 → 𝑉 ⊗𝐾 𝑊 is universal in the category ℳℒ(𝑉, 𝑊) of all
middle linear maps on 𝑉 × 𝑊.
Reference(s):
1. Ameri R., On categories of hypergroups and hypermodules, Journal of Discrete
Mathematical Sciences and Cryptography, 6, 2-3, 121-132, 2003.
2. Ameri R., Dehghan O.R., On dimension of hypervector spaces, European Journal of Pure
and Applied Mathematics, 1, 2, 32-50,2008.
3. Ameri R., Borzooei R.A., Ghadimi K., Representations of polygroups, Italian Journal of
Pure and Applied Mathematics, 37, 595-610, 2016.
4. Ameri R., Borzooei R.A., Ghadimi K., Multivalued linear transformations of
hyperspaces, Ratio Mathematica, 27, 37-47,2014.
5. Ameri R., Ghadimi K., Borzooei R.A., Multivalued linear transformations in categories of
hypervector spaces,(submitted).
6. Corsini P., Prolegomena of Hypergroup Theory, Second Edition, Aviani Editor, 1993.
7. Corsini P., Leoreanu-Fotea V., Applications of Hyperstructure Theory, Kluwer Academic
Publishers, Dordrecht, Hardbound, 2003.
8. Davvaz B., Polygroup Theory and Related Systems, World Scientific, 2013.
9. Davvaz B., Leoreanu-Fotea V., Hyperring Theory and Applications, International
Academic Press, USA, 2007.
10. Marty F., Sur une g ́en ́eralisation de la notion de groupe. In 8`eme congr`es des
Math ́ematiciens Scandinaves, Stockholm , 45-49, 1934.
11. Motameni M., Ameri R., Sadeghi R., Hypermatrix based on Krasner hypervector spaces,
Ratio Mathematica, 25, 77-94,2013.
12. Vougiouklis T., Hyperstructures and Their Representations, Hadronic Press, Inc., 115,
Palm Harber, USA, 1994.
56
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili
E-mail(s): Jafar A’zami, [email protected], [email protected]
Fuzzy coprimary submodules and their representation
Jafar A'zami1
Let R be a commutative ring with non-zero identity and let M be a non-zero unitary R-
module. The concept of fuzzy coprimary submodule as a dual notion of fuzzy primary will be
studied. Among other things, the behavior of this concept with respect to fuzzy localization
formation and fuzzy quotient will be examined. Also the uniqueness theorem for a non-zero
fuzzy representable submodule of M will be proved.
Keywords: Fuzzy coprimary submodule, Fuzzy prime and primary ideal, Fuzzy
localization, fuzzy coprimary representation, fuzzy attached primes.
2010 AMS Classification: 08A72
Reference(s):
1. Kirby D. Coprimary decomposition of Artinian modules,J. London Math. Soc, 6, 571–
576, 1973.
2. Macdonald I. G., Secondary representation of modules over a commutative ring, Symposia.
Math. 11, 23–43, 1973.
3. Mordeson J. N., and Malik D. S., Fuzzy Commutative Algebra, World Scientific, 1998.
57
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1AmasyaUniversity, Department of Mathematics, Amasya, Turkey
E-mail: [email protected]
Constructing Topological Hyperspace with Soft Sets
Güzide Şenel1
In this study, by defining soft ditopological spaces, I construct a topological
hyperspace with soft sets. I make a new approach to the soft topology via soft set theory, with
defining two structures on a soft set - a soft topology and a soft subspace topology. Moreover,
I characterize separation axioms in soft ditopological spaces and investigate the relations
between soft topological and soft ditopological structures [10]. Based on this idea, the
relations between the separation axioms of ordered soft topological spaces and the separation
axioms of the corresponding soft ditopological spaces are established.
Keywords: Soft sets, hyperspace, topological hyperspace
2010 AMS Classification: 54B20
References:
1. Ali, M.I., Feng, F., Liu, X., Min, W.K., Shabir, M., On some new operations in soft set theory,
Computers and Mathematics with Applications 57, 1547-1553, 2009.
2. Aygünoglu, A. Aygün, H., Some notes on soft topological spaces, Neural Computation and
Application, 2011.
3. Aktaş, H. ve Cagman, N., Soft sets and soft groups, Information Sciences, 177(1), 2726-2735, 2007.
4. Cagman, N., Enginoglu, S., Soft set theory and uni-int decision making, European Journal of
Operational Research 207, 848-855, 2010.
5. Dizman, T. et al., Soft Ditopological Spaces, Filomat 30:1, 209–222, 2016.
6. Ittanagi, B.M., Soft Bitopological Spaces, International Journal of Computer Applications
(0975 8887), Volume 107 - No. 7, December 2014.
7. Maji, P.K., Biswas, R., Roy, A.R., An Application of Soft Sets in A Decision Making Problem,
Computers and Mathematics with Applications 44, 1077- 1083, 2002.
8. Maji, P.K., Biswas, R., Roy, A.R., Soft set theory, Computers and Mathematics with Applications
45, 555-562, 2003.
9. Molodtsov, D.A., Soft set theory-first results, Computers and Mathematics
with Applications 37, 19-31, 1999.
10. Şenel, G., The Theory of Soft Ditopological Spaces, International Journal of Computer
Applications (0975 - 8887), Volume 150 - No.4, September 2016.
11. Şenel, G., Çağman, N., Soft Topological Subspaces, Annals of Fuzzy Mathematics and
Informatics, vol.10, no : 4, 525 - 535, 2015.
12. Shabir, M., Naz, M., On soft topological spaces, Computers and Mathematics with Applications
61, 1786-1799, 2011.
58
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails: [email protected], [email protected], [email protected] , [email protected]
Fuzzy Weakly Prime Γ-Ideal in Γ-Rings
Gülşah Yeşilkurt1, Serkan Onar
1, Deniz Sönmez
1 and Bayram Ali Ersoy
1
In this study, we investigate fuzzy weakly prime, fuzzy partial weakly prime and
fuzzy weakly semiprime Γ- ideal of a Γ-ring. We obtain some characterizations of fuzzy
weakly prime, fuzzy partial weakly and fuzzy weakly semiprime Γ-ideal of a Γ-ring. First we
give the definition of fuzzy weakly prime ideal, fuzzy weakly semiprime and fuzzy partial
weakly prime Γ- ideal. Further we give some properties of its.
Keywords: Fuzzy prime ideal, fuzzy weakly prime Γ-ideal, fuzzy partial weakly prime Γ-ideal,
fuzzy weakly semiprime Γ-ideal.
2010 AMS Classification: 03E72, 16D25, 16P99, 08A72.
References:
1. Mahjoob R., On Weakly Prime L-Ideals, Italian Journal of Pure and Applied Math., 36, 465-472,
2016
2. Zahedi, MM., A Characterization of L-Fuzzy Prime Ideals, Fuzzy Sets and Systems, 44 (1): 147-
160, 1991,
3. Rao, M. M. K., Fuzzy Prime Ideals in Ordered Γ-Semiring, Journal of International Mathemati-
cal Virtual Instutute, 7, 85–99, 2017
4. Mordeson, J.N. and Malik, D. S., Fuzzy Commutative Algebra, World Scientific Publishing Co.
Pte. Ltd., 1998
5. T. K. Dutta and T. Chanda, Fuzzy Prime Ideals in -rings, Bull. Malays. Math. Sci. Soc. (2) 30(1)
(2007), 65–73.
6. W. E. Barnes, On the -rings of Nobusawa, Pacific J. Math. 18(1966), 411–422.
7. T. K. Dutta and T. Chanda, Structures of fuzzy ideals of �-ring, Bull. Malays. Math. Sci.Soc. (2)
28(1)(2005), 9–18.
8. S. Kyuno, Prime ideals in gamma rings, Pacific J. Math. 98(2)(1982), 375–379.
9. Zadeh, L.A., Fuzzy Sets, Inform. and Control, 8 (1965), 338-353.
10.Anderson, D.D., Smith, E., Weakly prime ideals, Houston J. of Math., 29 (4)(2003), 831-840.
11. Ebrahimi Atani, Sh., Farzalipour, F., On weakly primary ideals, Georgian Mathematics Journal, 12
(3) (2005), 423-429.
59
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1YildizTechnicalUniversity,Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected] , [email protected] , [email protected] , [email protected]
Intuitionistic Fuzzy 2-Absorbing Ideals of Commutative Rings
Sanem YAVUZ1, Serkan ONAR
1, Deniz SONMEZ
1, Bayram Ali ERSOY
1
The aim of this paper is to give a definitions of intuitionistic fuzzy 2- absorbing ideals
and intuitionistic fuzzy weakly completely 2- absorbing ideals of commutative rings and to
give their properties. Moreover, we give diagram which transition between definitions of
intuitionistic fuzzy 2- absorbing ideals of commutative rings.
Keywords: Fuzzy Set, Intuitionistic Fuzzy Set, Intuitionistic Fuzzy Ideal, Intuitionistic
Fuzzy Prime Ideal, 2-Absorbing Ideal, Intuitionistic Fuzzy Completely Prime Ideal,
Intuitionistic Fuzzy Weakly Completely Prime Ideal, Intuitionistic Fuzzy K- Prime Ideal,
Intuitionistic Fuzzy 2-Absorbing Ideal, Intuitionistic Fuzzy Strongly 2-Absorbing Ideal,
Intuitionistic Fuzzy Weakly Completely 2- Absorbing Ideal, Intuitionistic Fuzzy K-2-
Absorbing Ideal.
2010 AMS Classification: 08A72
References:
1. Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.
2. Atanassov K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96, 1986.
3. Hur K., Kang H.W. and.Song H.K, Intuitionistic fuzzy subgroups and subrings, Honam
Math J., 25, 19-41 , 2003.
4. Marashdeh M.F., Salleh A.R., Intuitionistic fuzzy rings, International Journal of Algebra, 5,
37-47 , 2011.
5. Badavi A., On 2- absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75,
417-429 , 2007.
6. Darani A.Y., On L- fuzzy 2- absorbing ideals, Italian J. of Pure and Appl. Math., 36, 147-
154 , 2016.
7. Bakhadach I., Melliani S., Oukessou M. and Chadli L.S., Intuitionistic fuzzy ideal and
intuitionistic fuzzy prime ideal in aring, Intuitionistic Fuzzy Sets (ICIFSTA), 22, 59-63, 2016.
8. Malik D.S. and Mordeson J.N., Fuzzy Commutative Algebra, World Scientific Publishing,
1998.
60
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematic Education, Istanbul, Turkey 2Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail: [email protected] , [email protected]
Transition from Two-Person Zero-Sum Games to Cooperative Games with Fuzzy
Payoffs
Adem C. Cevikel 1 and Mehmet Ahlatcioglu
2
In this study, we deal with games with fuzzy payoffs. We proved that players who are
playing a zero-sum game with fuzzy payoffs against nature are able to increase their joint
payoff, and hence their individual payoffs by cooperating. It is shown that, a cooperative
game with the fuzzy characteristic function can be constructed via the optimal game values of
the zero-sum games with fuzzy payoffs against nature at which players' combine their
strategies and act like a single player. It is also proven that, the fuzzy characteristic function
that is constructed in this way satisfies the superadditivity condition. Thus we considered a
transition from two-person zero-sum games with fuzzy payoffs to cooperative games with
fuzzy payoffs. The fair allocation of the maximum payoff (game value) of this cooperative
game among players is done using the Shapley vector.
Keywords: Cooperative game; Fuzzy number; Fuzzy games; Shapley vector.
2010 AMS Classification: 91A10, 91A12, 03E72.
References:
1. L.A.Zadeh, Fuzzy sets, Information and control, 8, (1965), 338-353.
2.H.J.Zimmermann, Fuzyy sets, Decision making, and expert systems, Kluwer academic publishers,
Boston, (1991).
3. D.Butnariu, Fuzzy games: A description of the concept, Fuzzy Sets and Systems 1 (1978) 181-192.
4. L.Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems
32 (1989) 275-289.
5. Nishizaki and M.Sakawa, Equilibrium Solutions in Multiobjective Bimatrix Games with Fuzzy
Payoffs and Fuzzy Goals, Fuzzy Sets and Systems, 111 (2000) 99-116.
6. M.Sakawa and I. Nishizaki, A solution concept in multiobjective matrix game with fuzzy payoffs
and fuzzy goals, Fuzzy logic and its applications to engineering, Information science, (1995), 417-426.
7. A.C.Cevikel and M.Ahlatçıoğlu, Solutions for fuzzy matrix games, Computers & Mathematics with
Applications, Vol.60,3, (2010), 399-410.
8. D.F.Li, A fuzzy multiobjective approach to solve fuzzy matrix games, The journal of fuzzy
mathematics, 7, (1999), 907-912.
9. D.F.Li, Fuzzy constrained matrix game with fuzzy payoffs, The journal of fuzzy mathematics, 7,
(1999), 873-880.
10. M.Sakawa and I. Nishizaki, Max-min solutions for fuzzy multiobjective matrix games, Fuzzy Sets
and Systems, 67 (1994) 53-69.
61
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, University of Mazandaran, Babolsar, Iran 2School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, iran 3Department of Mathematics, University of Mazandaran, Babolsar, Iran
E-mails: [email protected], [email protected], [email protected]
On computation of fundamental group of a finite hypergroup
K. Abbasi1, R. Ameri
2, Y. Talebi-Rostami
3
The purpose of this paper is the computation of fundamental group in a finite
hypergroup ).,( H In this regards we first obtain an algorithm to construct the equivalence
classes of ,* the fundamental relation on ,H then we construct ),,/( * H the fundamental
group of .H In particular, given some classes of hypergroups, we find fundamental groups of
them. We apply a comprehensive Java program to compute the fundamental group of a given
finite hypergroup. It consists of two sub-programs (Hypergroup generator and Main).
hypergroup generator, counts all hypergroups of order )(3 Nnn and isomorphism classes
of them and enumerates quasihypergroups of order n and all -equivalence classes and by
the next sub-program (Main), it is checked that is an arbitrary hypergroupoid ),( H of order
)( Nnn is a hypergroup or not. If it is a hypergroup, this sub-program computes its -
equivalence classes and fundamental group.
Keywords: Hypergroup, Fundamental relation, Fundamental group, Computation.
2010 AMS Classification: 20N20, 68W30, 68W40.
Acknowledgements:
The author partially has been supported by "Algebraic Hyperstructure Excellence (AHETM),
Tarbiat Modares University, Tehran, Iran" and "Research Center in Algebraic hyperstructures
and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran".
References:
[1] Ameri R., Nouzari T., A new characterization of fundamental relation on hyperrings,
Int. J. Contemp. Sciences, 5(10), 721-738, 2010.
[2] Bayon R., Lygeros N., Advanced results in enumeration of hyperstructures, Journal of
Algebra, 320, 821-835, 2008.
[3] Corsini P., Prolegomena of Hypergroup Theory, second edition., Aviani Editor, 1993.
[4]Corsini P., Leoreanu V., Applications of Hyperstructures Theory, Advanced in
Mathematics, Kluwer Academic Publishers, 2003.
62
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, University of Mazandaran, Babolsar, Iran 2School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, iran 3Department of Mathematics, University of Mazandaran, Babolsar, Iran
E-mails: [email protected], [email protected], [email protected]
[5] Freni D., A new characterization of the derived hypergroup via strongly regular
equivalences,,Communication in algebra, 30(8), 3977-3989, 2002.
[6] Freni D., On a Strongly Regular Relation in Hypergroupoids, Pure Math. Appl, Ser. A, 3-4
191-198, 1992.
[7] Freni D., Une note sur le coeur d'un hypergroupe et sur la cloture transitive ,* de , A
note on the core of a hypergroup and the transitive closure * of , Rivista. di Mat. Pura Appl
8, 153-156, 1991.
[8] Koskas M., Groupoides, Demi-hypergroupes et hypergroupes, J. Math. Pures Appl, 49,
155-192, 1970.
[9] Marty F., Sur une generalization de la notion de groupe, in th8 Congress
Math. Scandinaves, Stockholm, Sweden, 45-49, 1934.
[10] Vougiouklis T., The fundamental relations in hyperrings. The general hyperfield,
Proceeding of th4 International congress in Algebraic Hyperstructures and Its Applications
AHA, 1990.
63
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran
E-mails: [email protected], [email protected]
Various Kinds of Quotient of a Canonical Hypergroup
Hossein Shojaei1 and Reza Ameri
1
In this study various kinds of quotients of a given canonical hypergroup are introduced
and studied. In this regards we introduce some regular equivalence relations on a canonical
hypergroup to construct different quotients for this hyperstructure. We will proceed to
investigate the relationships among these relations such that these extracted quotient
structures be equal. Also, the relationship between the heart of a give canonical hypergroup
and its quotient via an equivalence relation is studied and some related basic results are
obtained. Finally, we study the quotient hyperstructures of a canonical hypergroup induced
via a normal canonical subhypergroup, and show that for this special kind of quotient space
all various kinds of quotients are concid.
Keywords: Hypergroup, Canonical Hypergroup, Quotient Hypergroup, Heart.
2010 AMS Classification: 20N20, 20N15
References:
1. R. Ameri, On categories of hypergroups and hypermodules, Journal of Discrete
Mathematical Sciences and Cryptography, 6(2-3) (2003), 121--132.
2. P. Corsini, Prolegomena of Hypergroup Theory, Second edition, Aviani editore, Tricesimo,
1993.
3. P. Corsini and V. Leoreanu, Applications of Hyperstructures Theory, Advanced in
Mathematics, Kluwer Academic Publisher, Dordrecht, 2003.
4. B. Davvaz, Polygroup Theory and Related Systems, World Scienti c, 2013.
5. B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International
Academic Press, USA, 2007.
6. D. Freni,Une note sur le cur d'un hypergroupe et sur la cloture transitive, Riv. Mat. Pura
Appl. 8 (1991), 153-156.
7. F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandenaves,
Stockholm, (1934), 45-49.
64
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran
E-mails: [email protected], [email protected]
8. M. Mittas, Hypergroupes canoniques, Math. Balkanica, 2, (1972), 165-179.
9. H. Shojaei, R. Ameri, On hypergroups with trivial fundamental group, in: 46th Annual
Iranian Mathematics Conference, Yazd University, (2015), 238-241.
10. M. Velrajan, A. Arjunan, Note on isomorphism theorems of hyperrings, Int. J. Math. and
Math. Sci (2010), Article ID 376985.
11. T. Vougiouklis, Hyperstructures and their Representations, Hadronic
Press, Inc., Palm Harber, USA, 1994.
65
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Ege University, Institute of Science, Department of Mathematics, Izmir, Turkey )
2Ege University, Department of Mathematics, Izmir, Turkey)
E-mails: [email protected], [email protected]
On multipliers of hyper BCC-algebras
Didem SÜRGEVİL UZAY1, Alev FIRAT
2
In this paper, we introduced the notion of multiplier of a hyper BCC-algebra, and
investigated some properties of hyper BCC- algebras. And then we introduced notion of
kernels. Also we gave some propositions related with isotone and Fixd(H).
Keywords: hyper BCC-algebra, multiplier, isotone, Fixd(H), regular.
2010 AMS Classification: 20N20, 16W25
Reference(s):
1. Marty F., Surene generalization de la notion de group, In eigth Congress Math.,
Scandinaves, Stockholm, 45-49, 1934.
2. Iseki K., An algebra related with propositional calculus, On hyper BCK algebras,
Italian Journal of Pure and Applied Msthematics, 8, 127-136, 2000.
3. Komori Y., The class of BCC-algebras is not a variety, Mathematica Japonica
29(3), 391-394, 1984.
4. Jun Y.B., Zahedi M.M., Xin X. L. Borzooei R.A., On hyper BCK-algebras,
Italian Journal of Pure and Applied Mathematics, 8, 127-136, 2000.
5. . Jun Y.B., Xin X. L., Zahedi M.M and Roh E.H., Strong hyper BCK-ideals of hyper
BCK-algebras,Math. Japonica 51(3), 493-498, 2000.
6. . Borzooei R.A., Dudek, W.A., Koohestani N., On hyper BCC-algebras, Hindawi
Publishing Corporation, International Journal of Math. Sci., 1-18, 2006.
7. Kim K.H., Lim H.J., On Multipliers of BCC-algebras, Honam Mathematical Jour-
nal J. 35(2), 201-210, 2013.
66
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yaşar University, Department of Mathematics, Izmir, Turkey
2Ege University, Faculty of Science, Department of Mathematics,Izmir, Turkey
E-mails:[email protected], [email protected]
Derivations on Hyperlattices
Şule Ayar ÖZBAL1, Alev FIRAT
2
In this paper, we introduced the notion of derivations on hyperlattices and investigated
some related properties. Also, we characterized the Fixd(L), and Kerd(L) by derivations
Keyword(s): Lattices, hyperlattices, derivations.
2010 AMS Classification: 03G10, 20N20, 16W25
References:
1. Yon Y.H. , Kim K.H., On Expansive Linear Maps and V-multipliers of Lattices ,
Quaestiones Mathematicae, 33:4, 417-427.
2. Szasz G., Derivations of Lattices, Acta Sci. Math. (Szeged) 37 (1975), 149-154.
3. Szasz G., Translationen der Verbande, Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961),
53-57.
4. Szaz A., Partial Multipliers on Partiall Ordered Sets, Novi Sad J. Math. 32(1) (2002), 25-
45.
5. Szaz A. And Turi J. , Characterizations of Injective Multipliers on Partially Ordered Sets,
Studia Univ. "BABE-BOLYAI" Mathematica XLVII(1) (2002), 105-118.
6. Posner E.C., Derivations in prime rings, Proc. Amer. Math. Soc. 8,(1957), 1093-1100.
7. Marty F., Surene generalization de la notion de group, In eigth Congress Math.,
Scandinaves, Stockholm, (1934), 45-49.
8. James J., Transposition: Noncommutative Join Spaces, Journal of Algebra, 187(1997),97-
119.
9. Rosaria R., Hypera ne planes over hyperrings, Discrete Mathematics, 155,(1996),215-223.
10. Xin X. L., BCI-algebras, Disccussion Mathematicae General Algebra and Applications,
26(2006),5-19.
11. Rasouli S. and Davvaz B., Lattices Derived from Hyperlattices, Communivations in
Algebra, 38:8, (2010), 2720-2737.
12. Ozbal S. A, Firat A. (2010). Symmetric Bi-Derivations of Lattices. Ars Combinatoria,
97(4), 471-477.
13. Ferrari L., On Derivations of Lattices, Pure Math. Appl. 12 (2001), no.4, 365-382.
14. Xin X. L., Li T.Y. and Lu J. H., On derivations of Lattices, Information Sciences 178,
(2008) 307-316.
67
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1University of Mohaghegh Ardabili, Ardabil, Iran
E-mail: [email protected]
On fuzzy -prime ideals
Naser Zamani1
Let R be a commutative ring with identity. Let FI(R) be the set of all fuzzy ideals of R
and : 0RFI R FI R be a function. We introduce the concept of fuzzy -prime
ideals. Some relationships between fuzzy -prime ideals and prime ideals of R will be
investigated. We find conditions under which fuzzy -primness gives primness and vice
versa. Also, the behaviour of this concept in rings product will be studied.
Keywords: Fuzzy -prime ideals, fuzzy prime ideal
2010 AMS Classification: 08A72
References:
68
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, Faculty of Science, , Khon Kaen University, Khon Kaen 40002, Thailand
2Department of Mathematics, Yazd University,Yazd, Iran
E-mails: [email protected] , [email protected]
On pure hyperideals in ordered semihypergroups
Thawhat Changphas 1 and Bijan Davvaz 2
In this study, the notions of pure hyperideal, weakly pure hyperideal and purely prime
hyperideal in ordered semihypergroups are introduced and studied. We prove that the set of
all purely prime hyperideals is topologized.
Keywords: Algebraic hyperstructure, ordered semihypergroup, weakly regular, püre
hyperideal, weakly pure hyperideal, purely prime hyperideal, topology
2010 AMS Classification: 20N20
References:
69
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran
2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran. 3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran
E-mail(s): [email protected] , [email protected], [email protected]
Relation Between Hyper K-algebras and Superlattice
(Hypersemilattice)
A.Rezazadeh1, A. Radfar
2, R. A. Borzooei
3
In this paper, by considering the notions of hypersemilattice and superlattice, we prove
that any commutative and positive imolicative hyper K-algebra, is a hypersemilattice.
Moreover, we prove that any bounded commutative hyper K-algebra with condition L is a
superlattice.
1. Introduction
The theory of hyperstructures was introduced in 1934 by Marty [5]. This theory has been
subsequently developed by the contribution of various authors. In [1], R. A. Borzooei et al.
applied the hyperstructures to K-algebras and introduced the notion of a hyper K-algebra and
investigated some related properties. Some researchers applied the hyperstructure to some
accepts of lattice theory and the notion of hypersemilattice was introduced by Z. Bin et al. in
[2] and the notion of superlattice was introduced by Mittas and Konstantinidou in [6]. In this
paper, we prove that every hyper K-algebra by some condition is a hypersemilattice. In
follow, we introduce the notions ∧ and ∨ on hyper K-algebras and we prove that every hyper
K-algebra
of order 3 by some condition is a superlattice.
2. Preliminary
In this section, we give some definitions and theorems that we need in the next sections.
Definition 2.1. [2] Let L be a nonempty set with a binary hyperoperation ⊗ on L such that for
all a, b, c ∈ L,
the following condition hold:
(i) a ∈ a ⊗ a,
(ii) a ⊗ b = b ⊗ a,
(iii) (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c).
Then (L, ⊗) is called a hypersemilattice.
Definition 2.2. [6] A superlattice is a partially ordered set (S, <) with two hyper operations ∨
and ∧ such that the following properties hold:
(S1) a ∈ a ∨ a and a ∈ a ∧ a,
(S2) a ∨ b = b ∨ a and a ∧ b = b ∧ a
70
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran
2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran. 3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran
E-mail(s): [email protected] , [email protected], [email protected]
(S3) (a ∨ b) ∨ c = a ∨ (b ∨ c) and (a ∧ b) ∧ c = a ∧ (b ∧ c),
(S4) a ∈ a ∨ (a ∧ b) and a ∈ a ∧ (a ∨ b),
(S5) if a < b then (b ∈ a ∨ b and a ∈ a ∧ b),
(S6) (b ∈ a ∨ b or a ∈ a ∧ b) implies a < b.
for all a, b, c ∈ S
Definition 2.3. [1] By a hyper K-algebra we mean a nonempty set H endowed with a
hyperoperation ”◦” and a constant 0 satisfy the following axioms:
(HK1) (x ◦ z) ◦ (y ◦ z) < x ◦ y,
(HK2) (x ◦ y) ◦ z = (x ◦ z) ◦ y,
(HK3) x < x,
(HK4) x < y and y < x imply x = y,
(HK5) 0 < x.
for all x, y, z ∈ H, where x < y is defined by 0 ∈ x ◦ y and for every A, B ⊆ H, A < B is defined
by ∃a ∈ A, ∃b ∈ B such that a< b.
Theorem 2.4. [1] Let H be a hyper K-algebra. Then the following are hold:
(i) x ∈ x ◦ 0,
(ii) x ◦ y < z ⇔ x ◦ z < y,
(iii) x ◦ (x ◦ y) < y,
(iv) x ◦ y < x,
(v) A ◦ B < A.
for all x, y, z ∈ H.
3. Relation between hyper K-algebras and hypersemilattice
In this section we prove that every commutative and positive implicative hyper K-algebra is a
hypersemilattice.
Definition 3.1. A hyper K-algebra (H, ◦,0) is called commutative if for all x, y ∈ H,
x ◦ (x ◦ y) = y ◦ (y ◦ x)
Notation. In any commutative hyper K-algebra, for all x, y ∈ H, we denote
x ∩ y = {z | z ∈ y ◦ (y ◦ x)}
Theorem 3.2. Let H be a commutative hyper K-algebra. Then we have the following
properties:
(i) x ∩ y < x and x ∩ y < y,
(ii) x ∩ y = y ∩ x,
71
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran
2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran. 3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran
E-mail(s): [email protected] , [email protected], [email protected]
(iii) x ∈ x ∩ x,
(iv) If x < y, then x ∈ x ∩ y.
for all x, y ∈ H.
Definition 3.3. A hyper K-algebra (H, ◦) is said to be positive implicative if for all x, y, z ∈ H,
(x ◦ y) ◦ z = (x ◦ z) ◦ (y ◦ z)
Theorem 3.4. Let H be a commutative and positive implicative hyper K-algebra. Then for all
x, y, z ∈ H,
(x ∩ y) ∩ z = x ∩ (y ∩ z)
Corollary 3.5. Let (H, ◦) be a commutative and positive implicative hyper K-algebra. Then
(H, ∩) is a hypersemilattice.
Example 3.6. Let H = {0, a, b} and the hyper operation ◦ is defined on H as follows:
o 0 a b
0 {0} {0,a} {0,a,b}
a {a} {0,a} {0,a,b}
b {b} {b} {0,a,b}
Then (H, ◦) is a commutative and positive implicative hyper K-algebra. Also we can see that
(H, ∩) is a hypersemilattice.
4. Relation between hyper K-algebras and superlattice
In this section we introduce the notion hypermeet ∧ on hyper K-algebras.
Definition 4.1. A hyper K-algebra (H, ◦,0) is called to be bounded, if there exist an element 1
such that x < 1, for all x ∈ H and is called complemented, if H is bounded and 1 ◦ x has a least
element with respect to <, for all x ∈ H.
We note that if H is bounded, then by (HK4) we can easily get that 1 is unique. Also if H be
complemented, then we use x' to denote min(1 ◦ x).
Notation: In any commutative hyper K-algebra, we denote
x∧ y = {z | z ∈ y◦(y◦x) s.t z < x and z <y},
for all x, y ∈ H.
Theorem 4.2. Let H be a commutative hyper K-algebra. If (x ∧ y) ∧ z = x ∧ (y ∧ z), for all x,
y, z ∈ X, then x ∧ y is a greatest lower bound of x and y.
72
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran
2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran. 3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran
E-mail(s): [email protected] , [email protected], [email protected]
Definition 4.3. Let H be a bounded commutative complemented hyper K-algebra. We say H
satisfies in conditions L, if for all x,y ∈ H,
(L1) x ∧ y ≠∅,
(L2) x' ◦ y = y' ◦ x,
(L3) (x')' = x.
Notation. In any bounded commutative hyper K-algebra with condition L, we define
x ∨ y = {z | z ∈ (x'∧ y')'}
for all x, y ∈ H.
Proposition 4.4. Let H be a bounded commutative hyper K-algebra with condition L. Then
the following hold:
(i) x < x ∨ y and y < x ∨ y,
(ii) x ∈ x ∨ x,
(iii) x ∨ y = y ∨ x,
(iv) If x < y, then y ∈ x ∨ y,
(v) If (x ∈ x ∧ y or y ∈ x ∨ y), then x < y,
for all x, y ∈ H.
Theorem 4.5. Let H be a bounded commutative hyper K-algebra with condition L. If (x∧y)∧z
= x∧(y∧z), then for all x, y, z ∈ H, (x ∨ y) ∨ z = x ∨ (y ∨ z)
Theorem 4.6. Let H be a bounded commutative hyper K-algebra with condition L and ∧ be
associative. Then x ∨ y is a lowest upper bound of x and y, for all x, y, z ∈ H.
Theorem 4.7. Let H be a bounded commutative hyper K-algebra with condition L. Then for
any x, y ∈ H, we have x ∈ x ∧ (x∨ y) and x ∈ x ∨ (x ∧ y).
Corollary 4.8. Let H be a bounded commutative hyper K-algebra with condition L and ∧ be
associative. Then (H, ∧, ∨) is a superlattice.
5. Bounded commutative hyper K-algebra of order 3 with condition L
In this section we prove that every hyper K-algebra of order 3 with condition L is a
superlattice.
Theorem 5.1. Let H = {0, a, 1} be a bounded commutative hyper K-algebra with condition L.
Then (x ∧ y) ∧ z = x ∧ (y ∧ z), for all x, y, z ∈ H.
73
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran
2 Department of Mathematics, Payame Noor University, p. o. box. 19395-3697, Tehran, Iran. 3 Department of Mathematics, Shahid Beheshti University, Tehran, Iran
E-mail(s): [email protected] , [email protected], [email protected]
Corollary 5.2. Let H = {0, a, 1} be a bounded commutative hyper K-algebra with condition L.
Then (H, ∧, ∨) is a superlattice.
In the next example we show that Theorem 5.1 is not correct for a hyper K-algebra of order
more than 3 in general.
Example 5.3. Let H = {0, a, b, 1} and the hyper operation ”◦” defined as follows:
Then (H, ◦) is a bounded commutative hyper K-algebra and satisfies in condition L. We can
see that ∧ is not associative operator.
Keyword(s): Hyper K-algebra, hypersemilattice, superlattice.
2010 AMS Classification: 03G10,06F35
Reference(s):
[1] R. A. Borzooei, A. Hasankhani and M. M. Zahedi, On hyper K-algebra, Math.Jpn., 52(2000), 113-
121.
[2] Z. Bin, X. Ying and H. S. Wei,Hypersemilattices, http://www.paper.edu.cn
[3] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publications
(2003).
[4] Y. B. Jun, M. M. Zahedi, X. L. Xin and R. A. Borzooei, On hyper BCK-algebras, Italian Journal of
Pure and Applied Mathematics, No. 10(2000), 127-136.
[5] F. Marty, Sur une generalization de la notion degroups, 8th Congress Math. Scandinaves,
Stockholm,
(1934), 45-49.
[6] J. Mittas and M. Konstantinidou, Sur une nouvelle generation de la notion de treillis. Les
supertreillis
et certaines de leurs proprites generales, Ann. Sci. Univ. Blaise Pascal, Ser. Math., vol.25, (1989), 61-
83.
[7] T. Roodbari, L. Torkzadeh and M. M. Zahedi, Simple hyper K-algebras, Quasigroups and Related
Systems, 16 (2008), 131-140.
o 0 a b 1
0 {0, a, b,
1}
{0, a, b,
1}
{0} {0}
a {a,b,1} {0, a, b,
1}
{0,a} {0}
b {b} {b} {0, a, b,
1}
{0, a, b,
1}
1 {1} {b} {a,b,1} {0, a, b,
1}
74
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Golestan University, Department of Mathematics, Gorgan, Iran
E-mail: [email protected]
VAGUE SOFT HYPERMODULES
T. Nozari 1
In this study, the notion of vague soft hypermodules as an extension of the notion of
vague soft hypergroups and vague soft hyperrings is introduced. Then some basic properties
of vague soft sets and homomorphisms between vague soft hypermodules are presented. Also
we studied the image and inverse image of a vague soft hypermodule under a vague soft
hypermodule homomorphism.
Keywords: Soft set, vague soft set, vague soft hypermodule.
2010 AMS Classification: 06D72 , 08A99, 20N20, 08A72.
References:
1. Ameri R, Nozari T, Fuzzy hyperalgebras, Comput. Math. Appl. 61, 2011, 149-154.
2. Ameri R, Norouzi M, H. Hedayati, Application of fuzzy sets and fuzzy soft sets
in hypermodules, RACSAM. 107, 327-338, 2013.
3. Bustince H, Burillo P, Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems. 79,
403-405, 1996.
4. Corsini P, Prolegomena of hypergroups theory, Second ed, Aviani Edotor, 1993.
5. Corsini P, Leoreanu V, Applications of hyperstructures theory, Advanced in Mathematics,
Kluwer Academic Publishers, 2003. SHORT TITLE OF THE PAPER SHOULD APPEAR
HERE 15
6. Gau W. L, Buehrer. D. J, Vague sets, IEEE Transactions on systems, Man and
Cybernetics. 23(2), 610-614, 1993.
7. Leoreanu-Fotea V, Fuzzy hypermodules,Comput. Math. Appl. 57, 466- 475, 2009.
8. Molodtsov D, Soft set theory- _rst results, Comput. Math. Appl. 37(4-5), 19-31,1999.
9. Rosenfeld A, Fuzzy groups, J. Math. Anal. Appl. 35, 512-517,1971.
10. Selvachandran G, SallehAlgebraic A. R, hyperstructures of vague soft sets associated with
hyperrings and hyperideals, Scienti_cWorldJournal. 2015.
11. Selvachandran G, SallehAlgebraic A. R, Vague soft hypergroups and vague
soft hypergroup homomorphism, Advances in Fuzzy Systems. 2014;2014:10.
doi: 10.1155/2014/758637.
12. Vougiouklis T, Hyperstructures and their representions, Hadronic Press Inc, Palm Harber,
1994.
13.. Xu W, Ma J, Wang S, Hao G, Vague soft sets and their properties, Comput.Math. Appl.
59, 787-794, 2010.
75
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
2 School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, Iran. 1,3 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.
E-mails: [email protected], [email protected], [email protected].
An introduction to Zero-Divisor Graph of a Commutative Multiplicative Hyperring
Z. Soltani 1 , R. Ameri 2 and Y. Talebi-Rostami 3
The purpose of this note is the study of zero-divisor graph of a commutative
multiplicative hyperrings, as a generalization of commutative rings. In this regards we consider
a commutative multiplicative hyperring (𝑅, +,∘), where (𝑅, +) is an abelian group, (𝑅,∘) is a
semihypergroup and for all 𝑎, 𝑏, 𝑐 ∈ 𝑅 , 𝑎 ∘ (𝑏 + 𝑐)⊆ (𝑎 ∘ 𝑏) + (𝑎 ∘ 𝑐) and (𝑎 + 𝑏) ∘ 𝑐 ⊆(𝑎 ∘
𝑐) + (𝑏 ∘ 𝑐). For 𝑎 ∈ 𝑅 a non-zero element 𝑏 ∈ 𝑅 is said to be a zero-divisor of a, if 0 ∈ 𝑎 ∘ 𝑏.
The set of zero-divisors of 𝑅 is denoted by 𝑍(𝑅). We associative to 𝑅 a zero-divisor graph
𝛤(𝑅), whose vertices of 𝛤(𝑅) are the elements of 𝑍(𝑅)∗ = 𝑍(𝑅)\{0} and two distinct vertices
of 𝛤(𝑅) are adjacent if they were in 𝑍(𝑅). Finally, we obtain some properties of 𝛤(𝑅) and
compare some of its properties to the zero-divisor graph of a classical commutative ring and
show that almost all properties of zero-divisor graphs of a commutative ring can be extend to
𝛤(𝑅) while 𝑅 is a strongly distributive multiplicative hyperring.
Keywords: Multiplicative hyperring, Zero-divisor graph, Fundamental relation.
AMS 2010 Classification: 20N20
References:
1. Ameri R., On the Categories of Hypergroups and Hypermodules, J. Discrete Math. Sci.
Cryptography 6, 121-132, 2003.
2. Anderson D.F., Badawi A., The total graph of a commutative ring, Journal of Algebra 320,
2706-2719, 2008.
3. Anderson D.F., Frazier A., Lauve A., Livingston P.S., The zero-divisor graph of a
commutative ring II: Lecture Notes in Pure and Appl. Math., vol. 220, Marcel Dekker, New
York, pp. 61-72, 2001.
4. Anderson D.F. , Livingston P.S., The zero-divisor graph of a commutative ring, J. Algebra,
217, 434-447,1999.
5. Beck I., Coloring of commutative rings, J. Algebra 116, 208-226, 1988.
6. Corsini P., Prolegomena of hypergroup theory, second edition Aviani, Editor, 1993.
7. Corsini P., V. Leoreanu, Applications of hyperstructure theory, Kluwer academic
publications 2003.
8. Davvaz B., Leoreanu V., Hyperrings theory and application, International Academic Press,
2007.
9. Vougiouklis T., Hyperstructures and their representations, Hardonic, press, Inc. 1994.
10. Zahedi M.M., Ameri R., On the Prime, primary and maximal subhypermodules, Italian.J.
Pure and Appl. Math., 5, 61-80, 1999.
76
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Niovi Kehayopulu(University of Athens, Department of Mathematics, 15784 Panepistimiopolis, Greece)
E-mail: [email protected]
Some ordered hypersemigroups which enter their properties into their σ-classes
Niovi Kehayopulu 1
We are interested in ordered hypersemigroups H which enter their properties into their
σ-classes, where σ is a complete semilattice congruence on H. This gives information about
the structure of ordered hypersemigroups referring to the decomposition of these
hypersemigroups into components of the same type. We prove the following:
Theorem 1. If H is a regular, left (resp. right) regular or intra-regular ordered hypersemigroup
and σ a complete semilattice congruence on H, then the σ-class (a)σ of H is, respectively, a
regular, left (resp. right) regular or intra-regular (ordered) subsemigroup of H for every aH.
As a consequence, if H is a completely regular ordered hypersemigroup and σ a complete
semilattice congruence on H, then the σ-class (a)σ is a completely regular subsemigroup of H
for every aH.
Theorem 2. If H is a left (resp. right) quasi-regular or semisimple ordered hypersemigroup
and σ a complete semilattice congruence on H, then the σ-class (a)σ of H is, respectively so.
Theorem 3. If H is a left (resp. right) simple ordered hypersemigroup and σ a complete
semilattice congruence on H, then (a)σ is a left (resp. right) simple subsemigroup of H for
every aH.
Theorem 4. If H is a simple ordered hypersemigroup and σ a complete semilattice
congruence on H, then (a)σ is a simple subsemigroup of H for every aH.
Theorem 5. If H is an archimedean or weakly commutative ordered hypersemigroup and σ a
complete semilattice congruence on H, then (a)σ is, respectively so.
The ``-part" of the theorems above being obvious, we get characterizations of the above
mentioned types of ordered hypersemigroups via their σ-classes, σ being a complete
semilattice congruence on H.
Keywords: ordered hypersemigroup, regular, left regular, intra-regular, left quasi-regular,
semisimple, left simple, simple, archimedean, weakly commutative, complete semilattice
congruence.
2010 AMS Classification: 20M99 (06F05)
77
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
Study of 𝚪-hyperrings by fuzzy hyperideals with respect to a 𝒕-norm
Krisanthi Naka1, Kostaq Hila
2, Serkan Onar
3, Bayram Ali Ersoy
4
In this paper, we inquire further into the properties on some kind fuzzy hyperideals
and we study the Γ-hyperrings via 𝑇-fuzzy hyperideals. By means of the use of a triangular
norm 𝑇, we define, characterize and study the 𝑇-fuzzy left and right hyperideals, 𝑇-fuzzy
quasi-hyperideal and bi-hyperideal in Γ -hyperrings and some related properties are
investigated. We compare fuzzy hyperideal to 𝑇-fuzzy hyperideals. We have shown that Γ-
hyperring is regular if and only if intersection of any 𝑇-fuzzy right hyperideal with 𝑇-fuzzy
left hyperideal is equal to its product. We introduce the notion of 𝑇-fuzzy quasi-hyperideal
and 𝑇-fuzzy bi-hyperideal. We discuss some of its properties. We have shown that the meet of
𝑇-fuzzy right and 𝑇-fuzzy left ideal is a 𝑇-fuzzy quasi hyperideal of a Γ-hyperring. We
characterize regular Γ-hyperring with 𝑇-fuzzy quasi-hyperideal and 𝑇-fuzzy bi-hyperideal.
We also introduce the 𝑇-(𝜆, 𝜇)-fuzzy bi-hyperideals in Γ-hyperrings and investigate some of
their properties.
Keyword(s): Γ-hyperrings, 𝑡-norm, 𝑇-fuzzy (resp. left, right) hyperideal, 𝑇-fuzzy quasi(bi)-
hyperideal, 𝑇-(𝜆, 𝜇)-fuzzy bi-hyperideal.
2010 AMS Classification: 16Y99, 16D25, 20N20, 08A72.
References:
[1] Anthony, J.M., Sherwood, H. Fuzzy groups redefined, J. Math. Anal. Appl. 69, 124-130,
1979.
[2] Ameri, R., Nozari, T. A new characterization of fundamental relation on hyperrings, Int.
J. Contemp. Math. Sci. Vol. 5, no. 13-16, 721-738, 2010.
[3] Ameri, R., Shafiiyan, N. Fuzzy prime and primary hyperideals in hyperrings, Adv. Fuzzy
Math. 2, 83-99, 2007.
[4] Ameri, R., Hedayati, H., Molaee, A. On fuzzy hyperideals in Γ-hyperrings, Iranian J.
Fuzzy Syst. Vol. 6, No. 2, 47-59, 2009.
[5] Asokkumar, A., Velrajan, M. Characterizations of regular hyperrings, Ital. J. Pure Appl.
Math. No. 22, 115-124, 2007.
[6] Asokkumar, A., Velrajan, M. Hyperring of matrices over a regular hyperring, Ital. J. Pure
Appl. Math. No. 23, 113-120, 2008.
[7] Barghi, A.R. A class of hyperrings, J. Discrete Math. Sci. Cryptogr. Vol. 6, no. 2-3, 227-
233, 2003.
[8] Barnes, W.E. On the Γ-rings of Nabusawa, Pacific Journal of Mathematics, 18(3), 411-
422, 1966.
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected], [email protected], [email protected]
78
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics,Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected], [email protected], [email protected]
[9] Corsini, P. Prolegomena of hypergroup theory, Supplement to Riv. Mat. Pura Appl.
Aviani Editore, Tricesimo, 1993. 215 pp. ISBN: 88-7772-025-5.
[10] Corsini, P., Leoreanu, V. Applications of hyperstructure theory, Advances in
Mathematics (Dordrecht), 5. Kluwer Academic Publishers, Dordrecht, 2003. xii+322
pp. ISBN: 1-4020-1222-5.
[11] Corsini, P. Hypergroupes reguliers et hypermodules, Vol. 20, 121-135, 1975.
[12] Dasic, V. Hypernear-rings. Algebraic Hyperstructures and Applications (Xanthi, 1990),
75-85, World Sci. Publ., Teaneck, NJ, 1991.
[13] Davvaz, B., Salasi, A. A realization of hyperrings, Commun. Algebra 34(12), 4389-
4400, 2006.
[14] Davvaz, B. Isomorphism theorems of hyperrings, Indian J. Pure Appl. Math. Vol. 35,
no. 3, 321-331, 2004.
[15] Davvaz, B., Leoreanu-Fotea, V. Hyperring theory and applications, International
Academic Press, Palm Harbor, Fla, USA, 2007.
[16] Davvaz, B., Cristea, I. Fuzzy algebraic hyperstructures. An introduction. Studies in
Fuzziness and Soft Computing 321. Cham: Springer (ISBN 978-3-319-14761-1/hbk;
978-3-319-14762-8/ebook). x, 242 p.(2015).
[17] Davvaz, B. Fuzzy hyperideals in semihypergroups, Italian J. Pure Appl. Math. 8, 67-74,
2000.
[18] Davvaz, B. Fuzzy hyperideals in ternary semihyperrings, Iranian J. Fuzzy Systems, 6,
21-36, 2009.
[19] Gontineac, V.M. On hypernear-rings and H-hypergroups, in Algebraic Hyperstructures
and Applications, pp. 171-179, Hadronic Press, Palm Harbor, Fla, USA, 1994.
[20] Hila, K., Abdullah, S. A study on intuitionistic fuzzy sets in Γ-semihypergroups, Journal
of Intelligent & Fuzzy Systems 26, 1695–1710, 2014.
[21] Klement, E.P., Mesiar, R., Pap, E. Triangular Norms. Trends in Logic-Studia Logica
Library, 8. Kluwer Academic Publishers, Dordrecht, 2000. xx+385 pp. ISBN:0-7923-
6416-3.
[22] Krasner, M. A class of hyperrings and hyperfields, Int. J. Math. Math. Sci. Vol. 6, no. 2,
307-311, 1983.
[23] Leoreanu-Fotea, V., Davvaz, B. Fuzzy hyperrings, Fuzzy Sets and Systems 160, 2366-
2378, 2009.
[24] Leoreanu-Fotea, V., Davvaz, B. Join n-spaces and lattices, Multiple Valued Logic Soft
Comput. 15, 2008.
[25] Leoreanu-Fotea, V., Davvaz, B. n-hypergroups and binary relations, Eur. J.
Combinatorics 29, 1207-1218, 2008.
[26] Mittas, J. Hypergroupes canoniques, Mathematica Balkanica, vol. 2, 165-179, 1972.
[27] Mittas, J. Hyperanneaux et certaines de leurs proprietes, vol. 269, pp. A623-A626, 1969.
[28] Marty, F. Sur une generalization de la notion de group, Proceedings of the 8th Congres
Math. Scandinaves, Stockholm, Sweden, (1934), 45-49.
[29] Massouros, C.G. Quasicanonical hypergroups. Algebraic Hyperstructures and
Applications (Xanthi, 1990), 129-136, World Sci. Publ., Teaneck, NJ, 1991.
[30] Mirvakili, S., Davvaz, B. Relations on Krasner (m, n)-hyperrings, Eur. J. Combinatorics
31, 790-802, 2010.
[31] Ostadhadi-Dehkordi, S., Davvaz, B. Ideal theory in Γ-semihyperrings, Iranian Journal of
Science & Technology (IJST) 37A3: 251-263, 2013.
79
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics,Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected], [email protected], [email protected]
[32] Prenowitz, W., Jantosciak, J. Join Geometries,Springer UTM, 1979.
[33] Pianskool, S., Hemakul, W., Chaopraknoi, S. On homomorphisms of some multiplicative
hyperrings, Southeast Asian Bull. Math. Vol. 32, no. 5, 951-958, 2008.
[34] Rota, R. Strongly distributive multiplicative hyperrings, Journal of Geometry, vol. 39,
no. 1-2, 130-138, 1990.
[35] Rosenfeld, A. Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.
[36] De Salvo, M. Hyperrings and hyperfields, Annales Scientifiques de lUniversite de
Clermont-Ferrand II, no. 22, 89-107, 1984.
[37] Sen, M.K., Dasgupta, U. Hypersemiring, Bull. Calcutta Math. Soc. Vol. 100, no. 2, 143-
156, 2008.
[38] Schweizer, B., Sklar, A. A Statistical Metric spaces, Pacific J. Math. 10, No.1, 313-334,
1960.
[39] Stratigopoulos, D. Certaines classes dhypercorps et dhyperanneaux in Hypergroups,
Other Multivalued Structures and Their Applications, pp. 105-110, University of Udine,
Udine, Italy, 1985.
[40] Vougiouklis, T. Hyperstructures and their representations, Hadronic Press Monographs
in Mathematics. Hadronic Press, Inc., Palm Harbor, FL, 1994. vi+180 pp. ISBN: 0-
911767-76-2.
[41] Vougiouklis, T. The fundamental relation in hyperrings. The general
hyperfield.Algebraic Hyperstructures and Applications (Xanthi, 1990), 203-211, World
Sci. Publ., Teaneck, NJ, 1991.
[42] Ma, X., Zhan, J., Leoreanu-Fotea, V. On (fuzzy) isomorphism theorems of Γ-hyperrings,
Comput. Math. Appl. 60, no. 9, 2594-2600, 2010.
[43] Zahedi, M.M., Ameri, R. On the prime, primary and maximal subhypermodules, Ital. J.
Pure Appl. Math. 5, 61-80, 1999.
[44] Zadeh, L.A. Fuzzy sets, Inform. Control 8, 338-353, 1965.
.
80
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
On an algebra of fuzzy 𝒎-ary semihypergroups
Krisanthi Naka1, Kostaq Hila
2, Serkan Onar
3, Bayram Ali Ersoy
4
In this paper we deals with the fuzzy 𝑚-ary semihypergroups, fuzzy hyperideals and
homomorphism theorems on 𝑚-ary semihypergroups and fuzzy 𝑚-ary semihypergroups. We
also, introduce and study some classes of fuzzy hyperideals that of pure fuzzy, weakly pure
fuzzy hyperideals in 𝑚-ary semihypergroups and some properties of them are investigated.
We identify those 𝑚-ary semihypergroups for which every fuzzy hyperideal is idempotent.
We also characterize the 𝑚-ary semihypergroups for which every fuzzy hyperideal is weakly
pure fuzzy.
Keywords: 𝑚-ary semihypergroup, pure (weakly pure) fuzzy hyperideal, regular (weakly
regular) 𝑚-ary semihypergroups.
2010 AMS Classification: 20N15, 03E72, 20N20 .
References:
[1] Corsini, P. Prolegomena of hypergroup theory, Second edition, Aviani editor, 1993.
[2] Corsini, P., Leoreanu, V. Applications of hyperstructure theory, Advances in
Mathematics, Kluwer Academic Publishers, Dordrecht, 2003.
[3] Davvaz, B., Leoreanu, V. Binary relations on ternary semihypergroups, Commun.
Algebra 38(10), 3621-3636, 2010.
[4] Davvaz, B., Dudek, W.A., Vougiouklis, T. A Generalization of n-ary algebraic systems,
Commun. Algebra 37, 1248-1263, 2009.
[5] Davvaz, B., Dudek, W.A., Mirvakili, S. Neutral elements, fundamental relations and n-
ary hypersemigroups, Int. J. Algebra Comput. 19 (4), 567-583, 2009.
[6] Davvaz, B., Vougiouklis, T. 𝑛-Ary hypergroups, Iranian J. Sci. Tech., Transaction A, 30
(A2), 165-174, 2006.
[7] Davvaz, B. Fuzzy hyperideals in semihypergroups, Italian J. Pure Appl. Math. 8, 67-74,
2000.
[8] Davvaz, B. A survey of fuzzy algebraic hyperstructures, Algebra Groups and Geometries,
Vol. 27(1), 37-62, 2010.
[9] Davvaz, B., Cristea, I. Fuzzy algebraic hyperstructures. An introduction. Studies in
Fuzziness and Soft Computing 321. Cham: Springer (ISBN 978-3-319-14761-1/hbk;
978-3-319-14762-8/ebook). x, 242 p.(2015).
[10] Davvaz, B. Some Results on Congruences on Semihypergroups, Bull. Malaysian Math.
Sc. Soc. (Second Series) 23, 53-58, 2000.
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s): [email protected] , [email protected] , [email protected] , [email protected]
81
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics,Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails: [email protected] , [email protected], [email protected] , [email protected]
[11] Ghadiri, M., Waphare, B.N., Davvaz, B. 𝑛-ary 𝐻𝑣 -structures, Southeast Asian Bull.
Math. 34, 243-255, 2010.
[12] Hila, K., Davvaz, B., Naka, K., Dine, J. Regularity in terms of Hyperideals, Chinese J.
Math., vol. 2013, Article ID 167037, 4 pages, 2013.
[13] Kuroki, N. Fuzzy bi-ideals in Semigroups, Comment. Math. Univ. St. Paul. 28, 17-21,
1979.
[14] Leoreanu-Fotea, V., Davvaz, B. 𝑛 -hypergroups and binary relations, European J.
Combinat., 29(5), 1207-1218, 2008.
[15] Leoreanu-Fotea, V. A new type of fuzzy n-ary hyperstructures, Inform. Sci. 179, 2710-
2718, 2009.
[16] Marty, F. Sur une generalization de la notion de group, 8th Congres Math. Scandinaves,
Stockholm, 45-49, 1934.
[17] Ostadhadi-Dehkordi, S. Semigroup derived from (Γ, N)-semihypergroups and T-functor,
Discussiones Mathematicae: General Algebra and Applications 35, 79-95, 2015.
[18] Rosenfeld, A. Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.
[19] Vougiouklis, T. Hyperstructures and their representations, Hadronic Press, Florida,
1994.
[20] Zadeh, L.A. Fuzzy sets, Inform. Control 8, 338-353, 1965.
82
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1,2 Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
E-maisl: [email protected] , [email protected]
On Annihilator in Pseudo BCI-algebras
Habib Harizavi 1 and Ali Bandari 2
In this paper, the concept of annihilator in a pseudo BCI-algebra is introduced and
some related properties are investigated. Some necessary and sufficient conditions for a
pseudo BCI-algebra to be semisimple are given. Moreover, it is proved that the annihilator of
a closed ideal A, denoted by *A , is the greatest closed pseudo BCI-ideal of X contained in the
BCK-part of X and satisfied * 0A A .
Keywords: pseudo BCI-algebra, pseudo BCI-ideal, annihilator, normal ideal
2010 AMS Classification: 08A99, 03B60
Acknowledgement:
Authors thank the Research Council of Shahid Chamran University of Ahvaz for its financial
support.
References:
83
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
The embedding of an ordered semihypergroup in terms of fuzzy sets
Krisanthi Naka1, Kostaq Hila
2, Serkan Onar
3, Bayram Ali Ersoy
4
In this paper we have investigated an embedding theorem of ordered semihypergroups in
terms of fuzzy sets. We prove that an ordered semihypergroup 𝑅 is embedded in the set 𝐹(𝑅) of all
fuzzy subsets of 𝑅, which is an poe-semigroup with the ordered relation and the multiplication and
addition defined in this paper.
Keywords: semihypergroup, ordered semihypergroup, fuzzy sets
2010 AMS Classification: 08A72, 20N20, 20N25, 06F05.
References:
[1] Bakhshi, M., Borzooei, R.A. Ordered polygroups, Ratio Math. 24, 31-40, 2013.
[2] Changphas, T., Davvaz, B. Properties of hyperideals in ordered semihypergroups, Ital. J.
Pure Appl. Math. 33, 425-432, 2014.
[3] Corsini, P. Prolegomena of Hypergroup Theory, Second edition, Aviani Editore, Italy,
1993.
[4] Corsini, P., Leoreanu, V. Applications of Hyperstructure Theory, Advances in
Mathematics, Kluwer Academic Publishers, Dordrecht, 2003.
[5] Chvalina, J. Commutative hypergroups in the sence of Marty and ordered sets,
Proceedings of the Summer School in General Algebra and Ordered Sets, Olomouck,
(1994), 19-30.
[8] Davvaz, B., Corsini, P., Changphas, T. Relationship between ordered semihypergroups
and ordered semigroups by using pseudoorder, European J. Combinatorics, 44, 208-217,
2015.
[7] Davvaz, B., Leoreanu-Fotea, V. Hyperring Theory and Applications, International
Academic Press, USA, 2007.
[8] Davvaz, B., Corsini, P., Changphas, T. Relationship between ordered semihypergroups
and ordered semigroups by using pseudoorder, European J. Combinatorics, 44, 208-
217, 2015.
[9] Davvaz, B. Fuzzy hyperideals in semihypergroups, Italian J. Pure Appl. Math. 8, 67-74,
2000.
[10] Davvaz, B. Fuzzy hyperideals in ternary semihyperrings, Iranian J. Fuzzy Systems, 6,
21-36, 2009.
[11] Davvaz, B., Cristea, I. Fuzzy algebraic hyperstructures. An introduction. Studies in
Fuzziness and Soft Computing 321. Cham: Springer (ISBN 978-3-319-14761-1/hbk;
978-3-319-14762-8/ebook). x, 242 p.(2015).
[12] Davvaz, B. A survey of fuzzy algebraic hyperstructures, Algebra Groups and
Geometries, Vol. 27(1), 37-62, 2010.
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails:[email protected], [email protected], [email protected], [email protected]
84
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 2 University of Gjirokastra, Department of Mathematics & Computer Science, Gjirokastra, Albania 3 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey 4 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails:[email protected], [email protected], [email protected], [email protected]
[13] Davvaz, B. Semihypergroup Theory, Amsterdam: Elsevier/Academic Press (ISBN 978-
0-12-809815-8/pbk; 978-0-12-809925-4/ebook). viii, 156 p. (2016).
[14] Heidari, D., Davvaz, B. On ordered hyperstructures, Politehn. Univ. Bucharest Sci. Bull.
Ser. A Appl. Math. Phys. 73(2), 85-96, 2011.
[15] Gu, Z., Tang, X. Ordered regular equivalence relations on ordered semihypergroups, J.
Algebra, 450, 384-397, 2016.
[16] Hort, D.A. A construction of hypergroups from ordered structures and their morphisms,
J. Discrete Math. Sci. Cryptogr. 6, 139-150, 2003.
[17] Kehayopulu, N., Tsingelis, M. The embedding of an ordered groupoid into a poe-
groupoid in terms of fuzzy sets, Inform. Sci. 152, 231-236, 2003.
[18] Kuroki, N. Fuzzy bi-ideals in Semigroups, Comment. Math. Univ. St. Paul. 28, 17-21,
1979.
[19] Marty, F. Sur une generalization de la notion de groupe, 8 𝑖𝑒𝑚 Congres Math.
Scandinaves, Stockholm, Sweden, 1934, 45-49.
[20] Pibaljommee, B., Davvaz, B. Characterizations of (fuzzy) bi-hyperideals in ordered
semihypergroups, J. Intell. Fuzzy Systems 28, 2141-2148, 2015.
[21] Tang, J., Davvaz, B., Luo, Y.F. Hyperfilters and fuzzy hyperfilters of ordered
semihypergroups, J. Intell. Fuzzy Systems 29(1), 75-84, 2015.
[22] Rosenfeld, A. Fuzzy groups, J. Math. Anal. Appl. 35, 512-517, 1971.
[23] Vougiouklis, T. Hyperstructures and Their Representations, Hadronic Press, Palm
Harbor, Florida, 1994.
[24] Xie, X.Y., Wang, L. An embedding theorem of hypersemigroups in terms of fuzzy sets,
2012 9th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD
2012).
[25] Zadeh, L.A. Fuzzy sets, Inform. Control 8, 338-353, 1965.
.
85
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails: [email protected] , [email protected]
On δ-Primary Hyperideals of Commutative Semihyperrings
Ashraf Abumghaiseeb 1 and Bayram Ali Ersoy
1
In this work, we study the mapping 𝛿 that assigns to each hyperideal 𝐼 of the commu-
tative semihyperring 𝑅, another hyperideal 𝛿(𝐼) of the same semihyperring. Also we intro-
duced the notation of 𝛿-zero divisor of commutative semihyperring and 𝛿-semidomainlike
semihyperring which is generalization to those in the semirings. Moreover we showed that if
𝛿 be a global hyperideal expansion then 𝐼 is 𝛿-primary if and only if 𝑍𝛿(𝑅 𝐼) ⊆ 𝛿({0𝑅 𝐼⁄ })⁄ .
Keywords: Semihyperring, hyperideal, 𝛿-primary, 𝛿-semidomailike semihyperring.
2010 AMS Classification: 20N99, 13A15
References:
1. M. Shabir, N. Mehmood, P. Corsini, “Semihyperrings Characterized By Their Hyperide-
als”. Italian Journal of Pure and Applied Mathematics, accepted in March 2010.
2. R. Ameri, H. Hedayati, “On k-hyperideals of semihyperrings”, Journal of Discrete Mathe-
matical Sciences & Cryptography 10, No. 1 (2007), 41-54.
3. S. Ebrahimi Atani, Z. Ebrahimi Sarvandi and M. Shajari Kohan, “On 𝛿-Primary Ideals of
Commutative Semirings”, Romanian Journal of Mathematics and Computer Science, 2013,
Volume 3, Issue 1, P.71-81.
4. B. Davvaz, V. Leoreanu-Fote , “ Hyperring Theory and Applications”, International Aca-
demic Press. 2007.
5. P. J. Allen, “A fundamental theorem of homomorphisms for semirings”, Proc. Amer. Math.
Soc., 21(1969), 412-416.
6. Z. Dongsheng “δ-primary ideal of commutative rings”, Kyunkpook Math. Journal 41, 17-
22, 2001.
86
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail: [email protected]
On the Vahlen Matrices
Mutlu Akar1
After we recall definition of Vahlen Matrices, give their some properties.
Keyword(s): Clifford Algebra, Clifford Matrix, Möbius Transformations.
2010 AMS Classification: 11E88, 15A66.
References:
1. Hile G. N. and Lounesto P., Matrix Representations of Clifford Algebras, Linear Algebra
and Its Applications, 128, 51-63, 1990.
2. Waterman P. L., Möbius Transformations in Several Dimensions, Advances in
Mathematics, 101, 87-113, 1993.
3. Ryan J., The Conformal Covariance of Huygen’s Principle-Type Integral Formulae in
Clifford Analysis, Clifford algebras and spinor structures, Math. Appl. 321, Kluwer Acad
Publ. Dordrecht, 301–310, 1995.
4. Lawson J., Clifford Algebras, Möbius Transformations, Vahlen Matrices and B-loops,
Commentationes Mathematicae Universitatis Carolinae, 51,2, 319–331, 2010.
87
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 LMAM Laboratory, University Mohamed Seddik Ben Yahia Jijel, Jijel 18000, Algeria.
E-mail: [email protected]
On the Baireness of function spaces
Abderrahmane Bouchair 1
Let X be a topological space and α a nonempty family of compact subsets of X. Let Cα
(X) denote the space of continuous real-valued functions on X equipped with the set open
topology. A subfamily β of α is called moving off α if, for each there is with
. We say that X has the Moving Off Property (MOP) with respect to α, if every
subfamily of α which moves off α contains an infinite subfamily which has a discrete open
expansion in X.
Recently, Bouchair and Kelaiaia [1] proved that, for X paracompact q-space then Cα (X) is
Baire if and only if each point of X has a neighborhood from α. In this work we prove that if
X is first countable, then Cα (X) is a Baire space if and only if X has the Moving Off Property
with respect to α.
Keyword(s): Baire space, function space, set open topology, topological game
2010 AMS Classification: 54C35
Reference(s):
1. A. Bouchair, S. Kelaiaia,Comparison of some set open topologies on C(X,Y), Top.
Appl. 178, 352-359, 2014.
2. R.A. McCoy, I. Ntantu, Topological properties of spaces of continuous functions,
Lecture Note in Math. 1315. Springer Verlag Germany, (1988).
88
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Department of mathematics, University of Mohamed Seddik Ben Yahia-Jijel, Jijel, 18000 Algeria.
E-mails: [email protected], [email protected]
Global asymptotic stability of a higher order difference equation
Farida Belhannache1, Nouressadat Touafek
2
In this work, we investigate the global behavior of positive solutions of the
difference equation
𝑥𝑛+1 =𝐴+𝐵𝑥𝑛−2𝑘−1
𝐶+𝐷 ∏ 𝑥𝑛−2𝑖
𝑚𝑖𝑘𝑖=𝑙
, 𝑛 = 0,1, …
with non-negative initial conditions, the parameters 𝐴, 𝐵 are non-negative real
numbers, 𝐶, 𝐷 are positive real numbers, 𝑘, 𝑙 are non-negative fixed integers and 𝑚𝑖,
𝑖 ∈ {𝑙, … , 𝑘} are positive fixed integers such that 𝑙 ≤ 𝑘.
Keyword(s): Difference equation, global behavior, oscillatory, boundedness
2010 AMS Classification: 39A10
Reference(s):
1. Abo-Zeid R., Global behavior of a higher order difference equation, Math. Solvaca.,
64, 4, 931-940, 2014.
2. Belhannache F, Touafek N, Abo-Zeid R, Dynamics of a third-order rational difference
equation, Bull. Math. Soc. Sci. Math. Roumanie., 59, 1, 13-22, 2016.
3. Elsayed E. M, On the dynamics of a higher-order rational recursive sequence,
Commun. Math. Anal., 12, 117-133, 2012.
89
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Department of Mathematics, University of Annaba, Algeria 2Department of Mathematics University of Souk Ahras, Algeria
E-mails: [email protected] , [email protected].
Weak -favorability of C(X) with a set open topology
Kelaiaia Smail1 and Harkat Lamia
2
Let C(X) be the set of all continuous real valued functions we endow it with a set open
topology with the help of a family of -compact subsets of X. We use a topological game to
study the weak -favorability of C(X).
Keywords: Function spaces, set-open topologies, uniform topologies, topological games,
-favorability, -compact.
2010 AMS Classification: 54C35.
References:
1.A. Bouchair and S. Kelaiaia, "Application des jeux topologiques à l'étude de C(X) muni
d’une topologie set-open", Revue des Sciences etTechnologies. Univ. Mentouri. Constantine.
A-No 20.17-20. (2003).
2. G. Choquet, "Lectures in analysis," Benjamin, New York, Amsterdam, 1969.
3. R. Egelking, "General Topology," Polish scientifc publishing, 1977.
4. G. Gruenhage, "Games, covering properties and Eberlein compacts," Top.Appl. 23(1986),
291-297.
5. S. Kelaiaia, "On a completeness property of C(X)," Int. J. Appl. Math.6(2001), 287-291.
6. S. E. Nokhrin and A. V. Osipov, "On the Coicidence of Set-Open and Uniform
Topologies," Proc. Steklov. Inst. Suppl. 3(2009), 184-191.
7. R. A. McCoy and I. Ntantu, "Topological Properties of Spaces of Continuous Functions,"
Lecture Note, Springer-Verlag, Berlin, (1988).
8. R. A. McCoy and I. Ntantu, "Countability properties of function spaces with set-open
topology," Top. Proc. 10(1985), 329-345.
9. R. telgarsky, "Topological games on the 50th anniversary of the BanachMazur game,"
Topology Proc. 10, 329-345. (1987).
10. J.C. Oxtoby, "The Banach-Mazur games and Banach category theorem," Contribution to
the theory of games, Vl. III Annals of Math.Studies 39. Princeton UniversityPress. Princeton,
(1957).
90
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1,2Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran.
E-mail(s): [email protected], [email protected]
Curves on Lightlike Cone in Minkowski Space
Nemat Abazari1, Alireza Sedaghatdoost
2
In the Minkowski space 𝐸1𝑛, the set of all lightlike vectors is called lightlike cone and
it is denoted by 𝑄𝑛−1 . In this paper we use Frenet orthonormal frame and asymptotic
orthonormal frame for study of curves on the lightlike cone 𝑄𝑛−1 in Minkowski space 𝐸1𝑛. We
study all lightlike and spacelike curves in𝑄𝑛−1. We classify all curves with constant cone
curvature in 𝑄4, 𝑄5 and 𝑄6. Also we give some relation between Frenet curvature and cone
curvature functions for a curve in 𝑄3.
Keyword(s): Asymptotic frame, cone curvature, lightlike cone, spacelike curve.
2010 AMS Classification: 53A35
Reference(s):
1. D.N. Kupeli , On null submanifolds in spacetime,Geometriae Dedicata 23 (1987), 33-51.
2. H. Liu, Curves in the lightlike cone. Contrib. Algebr. Geom. 45 (2004), 291-303.
3. H. Liu, Q. Meng, Represeentation formulas of curves in a two- and threeDimensional
lightlike cone, Results in Math. 59 (2011), 437-451.
4. R. L´opez, On differential geometry of curves and surfaces in LorentzianMinkowski
space, International Electronic Journal of Geometry. 7 (2014), 44-107.
5. B. O’Neill, Semi-Riemannian Geometry, Academic Press, NewYork 1983.
6. M. P. Torgaˇsev, E. Su´curovi´c, ˇ W-Curves in Minkowksi space-time, Novi SaJ.
Math. Vol 32, No 2, 2002, 55-65.
7. S. Yilmaz, M. Turgut, On the differential geometry of the curves in Minkowski
space-time I, Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 27, 1343-1349.
91
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1,2Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails: [email protected], [email protected]
*This research is supported by Yildiz Technical University Scientific Research Projects Coordination
Department Project Number: 2016-01-03-DOP01.
A class of LCD codes from group rings*
Mehmet E. Koroglu1, Bayram A. Ersoy
1
Linear codes with complementary duals (abbreviated LCD) are linear codes that they
trivially intersect with their duals [1]. Finding new LCD code families are of great importance
due to their wide range of applications. Unit derived group ring codes were given by Hurley
and Hurley in [2]. Group rings are a rich source of unit elements. So it is possible to obtain
many new code parameters. In this work, we provide a necessary and sufficient condition for
the unit derived group ring codes to be LCD.
Keywords: Linear codes, LCD codes, group rings
2010 AMS Classification: 94B05, 94B60, 08A99
References:
1. Massey, J. L. Linear codes with complementary duals, Discrete Math. 106, 337-342, 1992.
2. Hurley, P. and Hurley, T. Codes from zero-divisors and units in group rings, International
Journal of Information and Coding Theory 1, 57-87, 2009.
92
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 2Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 3Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey
E-mail(s): [email protected], [email protected], [email protected]
New Blocking Cryptography Models
Sümeyra UÇAR1, Nihal TAŞ
2, Nihal Yılmaz Özgür
3
In this talk we present two new coding and decoding methods using Fibonacci Q
matrices and R-matrices. Since our methods study with small numbers, we obtain quite easy
methods than the known methods in literature.
Keyword(s): Coding algorithm / decoding algorithm / Fibonacci Q matrix / R matrix
2010 AMS Classification: 68P30, 11B39, 1B37.
Reference(s):
1. Bruggles I. D., Hoggatt V. E. Jr., A Primer for the Fibonacci numbers-Part IV. Fibonacci
Q. 1 (4), 65-71, 1963.
2. Koshy, T., Fibonacci and Lucas numbers with applications, New York, NY: JohnWiley and
Sons, 2001.
3. Prasad, B., Coding theory on Lucas p numbers, Discrete Mathematics, Algorithms and
Applications 8 (4), 17 pages, 2016.
4. Stakhov, A. P., Fibonacci matrices, a generalization of the Cassini formula and a new
coding theory, Chaos, Solitons Fractals 30 (1), 56-66, 2006.
5. Taş, N., Uçar, S. and Özgür, N. Y., Pell coding and Pell decoding methods with some
applications, arXiv:1706.04377 [math.NT].
6. Taş, N., Uçar, S., Özgür, N. Y. and Kaymak, Ö. Ö., A new coding/decoding algorithm
using Fibonacci numbers, submitted for publication.
7. Uçar, S., Taş, N. and Özgür, N. Y., A new cryptography model via Fibonacci and Lucas
numbers, submitted for publication.
93
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 2Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 3Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey
E-mail(s): [email protected], [email protected], [email protected]
A New Coding Theory with Generalized Pell (p,i) – Numbers
Nihal Taş1, Sümeyra Uçar
2, Nihal Yılmaz Özgür
3
Recently, it has been introduced a new coding algorithm, called blocking algorithm,
using Fibonacci (resp. Lucas) numbers and a blocking method. In this study, we develop a
new coding and decoding method using the generalized Pell (p,i) – numbers. We give an
application of generalized Pell (p,i) – numbers to blocking algorithm.
Keyword(s): Coding theory / decoding theory / generalized Pell (p,i) – numbers / blocking
algorithm
2010 AMS Classification: 68P30, 14G50, 11T71, 11B39.
Reference(s):
1. Kılıç, E., The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial
representations, sums, Chaos, Solitons Fractals, 40, 2047-2063, 2009.
2. Koshy, T., Pell and Pell-Lucas numbers with applications, Springer, Berlin, 2014.
3. Prasad, B., Coding theory on Lucas p numbers, Discrete Mathematics, Algorithms and
Applications 8 (4), 17 pages, 2016.
4. Stakhov, A. P., Fibonacci matrices, a generalization of the Cassini formula and a new
coding theory, Chaos, Solitons Fractals 30 (1), 56-66, 2006.
5. Taş, N., Uçar, S. and Özgür, N. Y., Pell coding and Pell decoding methods with some
applications, arXiv:1706.04377 [math.NT].
6. Taş, N., Uçar, S., Özgür, N. Y. and Kaymak, Ö. Ö., A new coding/decoding algorithm
using Fibonacci numbers, submitted for publication.
7. Uçar, S., Taş, N. and Özgür, N. Y., A new cryptography model via Fibonacci and Lucas
numbers, submitted for publication.
94
13th AlgebraicHyperstructuresandits Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 2Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 3Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey
E-mail(s):[email protected], [email protected], [email protected]
Some Problems in Spectral Graph Theory
Sezer Sorgun1, Hakan Küçük
2, Hatice Topcu
3
Spectral Graph Theory is the studies of Linear Algebra and Graph Theory mix
together. In this study, we usually find the relations between the eigenvalues of popular
matrices and the graph parameters. In this talk, we present some problems related to energy
problems, isomorphism etc. in the theory.
Keyword(s): Graph, Isomorphism, Energy, Graph Matrices, Eigenvalue
2010 AMS Classification: 05C50
Reference(s):
1. Haemers W., Bouwer A., Spectra of Graphs, Springer, 2010.
2. Haemers W., Dam E. R., Which Graphs are Determined by Their Spectrum?, Linear
Algebra Appl., 373, 241-272, 2003.
3. Li X., Shi Y., Gutman I., Graph Energy, Springer, New York, 2012.
4. Das K.C., Sorgun S., Gutman I., On Randić energy, MATCH Comm.Math.Comput. Chem.,
73, 81-92, 2015.
5. Das K.C., Sorgun S., On Randić energy of graphs, MATCH Comm.Math.Comput. Chem.,
72, 227-238, 2014.
6. Zhang X., Zhang H., Some Graphs Determined by Their Spectra, Linear Algebra Appl.
431,1443-1454, 2009.
7. Sorgun S., Topcu H., On the spectral characterization of kite graphs_ J. Algebra Comb.
Discrete Appl., 3 (2), 81–90, 2016.
8. Topcu H., Sorgun S., Haemers W., On the spectral characterization of pineapple graphs,
Linear Algebra and its Applications 507, 267–273, 2016.
95
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 2Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey 3Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevşehir Turkey
E-mail(s): [email protected], [email protected], [email protected]
On Trees Which Have Four Non-Zero Randić Eigenvalues
Hakan Küçük1, Sezer Sorgun
2, Hatice Topcu
3
A popular and important research field is to investigate the characterization of the
connected graphs with special and distinct eigenvalues. It is an interplay between
combinatorics and linear algebra. Moreover, Randić Matrix and Randić Energy studies in
Spectral Graph Theory are essential. In this presentation we give some basic information
about Randić Matrix, then we present our observations and conclusions about trees which
have four non-zero Randić eigenvalues.
Keyword(s): Graph, Matrices, Randić Matrix, Randić Eigenvalues, Trees
2010 AMS Classification: 05C50
Reference(s):
1. Diestel R., Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer-
Verlag, Heidelberg, 2010.
2. Chung F., Spectral Graph Theory, American Mathematical Society,National Science
Foundation, 1997.
3. Gu R. , Huang F. , Li X. , General Randić Matrix and General Randić Energy,
Transactions on Combinatorics, 3 (3) , 21-33, 2014.
4. Bozkurt Ş.B., Güngör A.D., Gutman I. , Çevik A. S., Randić Matrix and Randi¢
Energy, MATCH Commun. Math. Comput. Chem., 64 , 239-250, 2014.
5. Li X. , Wang J. , Randić Matrix and Randić Eigenvalues, MATCH Commun. Math.
Comput. Chem. 73, 73-80, 2015.
6. Dam E.R. , Graphs with few eigenvalues: An interplay between combinatorics and
algebra, PhD Thesis, Tilburg University, 1996.
96
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected]
One-Parameter Planar Motions in Generalized Complex Number Plane
Nurten (Bayrak) Gürses1 and Salim Yüce
1
The generalized complex numbers have the form
2, ( , ) where , ( , ).z x Jy x y J i q p p q
By taking 2 ; 0J p q and p , generalized complex number system can be
presented as follows:
2{ : , , }x Jy x y J p p .
p is called p -complex plane. Moreover, the set J is defined
2: , , , { 1,0,1}J x Jy x y J p p such that J ⊂ p . For p <0, p is called
elliptical complex, for p =0, p is called parabolic complex, and for p >0, p is called
hyperbolic complex number systems.
In this study, we firstly give the basic notations of the p -complex plane p . Then, we
introduce the one-parameter planar motions in p -complex plane J such that J ⊂ p .
These motions correspond the one-parameter motions in affine Cayley-Klein planes. We
examine this motion theory with aspects of complex motions. Besides, we discuss the
relations between absolute, relative, sliding velocities (accelerations) and pole curves under
the motions J / J .
Keywords: Generalized complex number plane, complex-type numbers, one-parameter
planar motion, kinematics.
2010 AMS Classification: 53A17, 53A35
References:
[1] N. (Bayrak) Gürses, S. Yüce, One-Parameter Planar Motions in Affine Cayley-Klein
Planes. European Journal of Pure and Applied Mathematics, 7 no. 3(2014), 335–342.
[2] W. Blaschke and H. R.Müller, Ebene Kinematik. Verlag Oldenbourg, München,1956.
[3] P. Fjelstad, Extending special relativity via the perplex numbers. Amer. J. Phys. 54 no.5
(1986), 416–422.
97
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected]
[4] D. Alfsmann, On families of 2N dimensional Hypercomplex Algebras suitable for digital
signal Processing. Proc. EURASIP 14th European Signal Processing Conference(EUSIPCO
2006), Florence, Italy, 2006.
[5] I.M. Yaglom, Complex numbers in geometry, Academic Press, New York, 1968.
[6] S. Yüce and N. Kuruoğlu, One-Parameter Plane Hyperbolic Motions. Adv. appl. Clifford
alg. 18 (2008), 279–285.
[7] P. Fjelstad and S.G. Gal, n-dimensional hyperbolic complex numbers. Adv. Appl. Clifford
Algebr. 8 no.1 (1998), 47–68.
[8] P. Fjelstad and S.G. Gal, Two-dimensional geometries, topologies, trigonometries and
physics generated by complex-type numbers. Adv. Appl. Clifford Algebra 11 no. 1 (2001)
81–107.
[9] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers.
Anal. Univ. Oradea, fasc. math. 11 (2004), 71–110.
[10] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, Hyperbolic Trigonometry in two-
dimensional space-time geometry. N. Cim. B 118 B (2003), 475491.
[11] E. Study, Geometrie der Dynamen. Verlag Teubner, Leipzig, 1903.
[12] F. M. Dimentberg, The Screw Calculus and Its Applications in Mechanics, Foreign
Technology Division translation FTD-HT-23-1632-67, (1965).
[13] F. M. Dimentberg, The method of screws and calculus of screws applied to the theory of
three dimensional mechanisms. Adv. in Mech. 1 no. 3-4 (1978), 91–106.
[14] Ö. Köse, Kinematic differential geometry of a rigid body in spatial motion using
dual vector calculus: Part-I. Applied Mathematics and Computation 183 no. 1 (2006), 17–29.
[15] G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous,
spatial kinematics. Mechanism and Machine Theory 11 no. 2 (1976), 141–156.
[16] A. A. Harkin and J. B. Harkin, Geometry of Generalized Complex Numbers.
Mathematics Magazine, 77 no. 2 (2014).
[17] F. Catoni, D. Boccaletti, , R. Cannata, V. Catoni, E. Nichelatti and P. Zampetti, The
mathematics of Minkowski space-time and an introduction to commutative hypercomplex
numbers. Birkh auser Verlag, Basel, 2008.
[18] I. M. Yaglom, A simple non-Euclidean geometry and its Physical Basis. Springer-Verlag,
New York, 1979.
[19] E. Pennestri and R. Stefanelli, Linear Algebra and Numerical Algorithms Using Dual
Numbers. Multibody System Dynamics, 18 no. 3 (2007), 323–344.
[20] G. Sobczyk, The Hyperbolic Number Plane. The College Math. J. 26 no. 4 (1995), 268–
280.
[21] F. Klein, Uber die sogenante nicht-Euklidische Geometrie. Gesammelte Mathematische
Abhandlungen, (1921), 254–305.
[22] F. Klein, Vorlesungen über nicht-Euklidische Geometrie. Springer, Berlin, 1928.
98
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected]
[23] H. Es, Motions and Nine Different Geometry. PhD Thesis, Ankara University Graduate
School of Natural and Applied Sciences, 2003.
[24] G. Helzer, Special Relativity with Acceleration. The American Mathematical Monthly,
107 no. 3 (2000), 219–237.
[25] F. J. Herranz and M. Santader, Homogeneous Phase Spaces: The Cayley-Klein
framework. http://arxiv.org/pdf/physics/9702030v1.pdf., (1997).
[26] R. Salgado, Space-Time Trigonometry. AAPT Topical Conference: Teaching General
Relativity to Undergraduates, AAPT Summer Meeting, Syrauce University, NY, July 20-
21,22-26, 2006.
[27] M. A. F. Sanjuan, Group Contraction and Nine Cayley-Klein Geometries. International
Journal of Theoretical Physics, 23(1) (1984).
[28] M. Spirova, Propellers in Affine Cayley-Klein Planes. Journal of Geometry, 93 (2009),
164–167.
[29] A. S. McRae, Clifford Fibrations and Possible Kinematics. Symmetry, Integrability and
Geometry: Methods and Applications, 5 (072) (2009).
[30] V. V. Kisil, Geometry of Möbius Transformations:Eliptic, Parabolic and Hyperbolic
Actions of SL2 (R). Imperial College Press, London, 2012.
[31] N. A. Gromov and S. S. Moskaliuk, Classication of transitions between groups in
Cayley-Klein spaces and kinematic groups. Hadronic J. 19 no. 4 (1996), 407–435.
[32] N. A. Gromov and V. V. Kuratov, Possible quantum kinematics. J. Math. Phys. 47 no. 1
(2006).
[33] A. A. Ergin, On the one-parameter Lorentzian motion. Communications, Faculty of
Science, University of Ankara, Series A 40 (1991), 59–66.
[34] M. Akar, S. Yüce and N. Kuruoğlu, One-Parameter Planar Motion in the Galilean Plane.
International Electronic Journal of Geometry (IEJG) 6 no. 1 (2013), 79–88.
[35] S. Yüce and M. Akar, Dual Plane and Kinematics. Chiang Mai J. Sci. 41 no.2 (2014),
463–469.
[36] H. R. Müller, Verallgemeinerung einer formel von Steiner. Abh. d. Brschw. Wiss. Ges.
24 (1978), 107–113.
[37] J. Hucks, Hyperbolic complex structures in physics. J. Math. Phys. 34 no. 12 (1993).
99
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Nevşehir Hacı Bektaş Veli University, Department of Mathematics, Nevsehir, Turkey,
2Ankara University, Department of Mathematics, ,Ankara, Turkey,
E-mails: [email protected] , [email protected]
A New Approach to Motions and Surfaces with Zero Curvatures in Lorentz 3-Space
Esma DEMİR ÇETİN 1 and Yusuf YAYLI
2
In this work we search for the surfaces with zero curvatures in Lorentz 3-space, whose
generating curve is a graph of a polynomial under homothetic motion groups. We study with
the generating curves α (s) = (f(s),0,g(s)), α (s) = (f(s),g(s),0), α (s) =(f(s),g(s),f(s)) depending
on the casual character of the axis. (Timelike axis, spacelike axis, lightlike axis respectively.)
First of all we see that the degree of the polynomials must be equal for zero curvatures. We
show that, distinct from the helicoidal motion groups, these surfaces generated by graph of
polynomials don’t have to be ruled surfaces for zero curvatures.
Keywords: Gauss curvature, mean curvature, umbilic points, Lorentz space, Homothetic
motion
2010 AMS Classification: 53B30, 53C50.
References:
100
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Balıkesir University, Department of Mathematics, Balıkesir, Turkey, 2 Balıkesir University, Department of Mathematics, Balıkesir, Turkey,
E-mails: [email protected], [email protected]
Chen-Ricci and Wintgen Inequalities for Statistical Submanifolds of Quasi-Constant
Curvature
Hülya Aytimur 1, Cihan Özgür
2
We define statistical manifold of quasi-constant curvature and give an example. We
find Chen-Ricci inequalities, generalized Wintgen inequality for submanifolds in a statistical
manifold of quasi-constant curvature.
Keywords: Statistical submanifold \ Chen-Ricci İnequalities \ Wintgen İnequality \ quasi-constant
curvature
2010 AMS Classification: 53C40, 53B05, 53B15, 53C05, 53A40
References:
1. S. Amari, Differential-Geometrical Methods in Statistics, Springer-Verlag, 1985.
2. M. E. Aydın, A. Mihai, I. Mihai, Some Inequalities on Submanifolds in Statistical
Manifolds of Constant Curvature, Filomat 29 (2015), no. 3, 465-477.
3. M. E. Aydın, A. Mihai, I. Mihai, Generalized Wintgen inequality for statistical
submanifolds in statistical manifolds of constant curvature, Bull. Math. Sci. (2016).
doi:10.1007/s13373-016-0086-1
4. P. W. Vos, Fundamental equations for statistical submanifolds with applications to the
Bartlett correction, Ann. Inst. Statist. Math. 41 (1989), no. 3, 429-450.
5. C. Özgür, B. Y. Chen inequalities for submanifolds of a Riemanian manifold of quasi-
constant curvature, Turk. J. Math. 35 (2011) 501-509.
6. B.-Y. Chen, Geometry of submanifolds. Pure and Applied Mathematics, No. 22. Marcel
Dekker, Inc., New York, 1973.
7. H. Furuhata, Hypersurfaces in statistical manifolds, Differential Geom. Appl. 27 (2009),
no. 3, 420-429.
101
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics,Istanbul,Turkey
2Yildiz Technical University, Department of Mathematics,Istanbul,Turkey
E-mails:[email protected], [email protected]
One-Parameter Homothetic Motion on the Galilean Plane
Mücahit Akbıyık1, Salim Yüce
2
In this paper, we will define one-parameter homothetic motion on the Galilean Plane.
The velocities, accelerations and pole points of the motion will be analysed.
Keywords: Galilean(Isotropic) Plane, Kinematics
2010 AMS Classification: 53A17
References:
1. Edmund Taylor Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid
Bodies, Cambridge University Press. Chapter 1, 1904.
2. Joseph Stiles Beggs, Kinematics, Taylor and Francis. p. 1, 1983.
3. Thomas Wallace Wright, Elements of Mechanics Including Kinematics, Kinetics and
Statics. E and FN Spon. Chapter 1, 1896.
4. A. Biewener, Animal Locomotion, Oxford University Press, 2003.
5. Blaschke, W., and Müller, H.R., Ebene Kinematik, Verlag Oldenbourg, München, 1959.
6. A. A. Ergin, On the one-parameter Lorentzian motion,Comm. Fac. Sci. Univ. Ankara,
Series A 40, 59-66, 1991.
7. Akar, M., Yüce, S. and Kuruoglu, N, One parameter planar motion in the Galilean plane,
International Electronic Journal of Geometry, 6(1), 79-88, 2013.
8. Hacisalihoğlu, H., On the rolling of one curve or surface upon another, Proceedings of the
Royal Irish Academy. Section A: Mathematical and Physical Sciences, Vol. 71, (1971), pp.
13-17.
9. Tutar, A., and Kuruoglu, N, On the one-parameter homothetic motions on the Lorentzian
plane, Bulletin of Pure & Applied Sciences, Vol. 18E, No.2 , 333-340, 1999.
10. Yaglom, I.M., A simple non-Euclidean Geometry and its Physical Basis, Springer-Verlag,
New York, 1979.
11. O. Röschel, Die Geometrie des Galileischen Raumes, Habilitationsschrift, Institut für
Math. und Angew. Geometrie, Leoben, 1984.
12. Helzer, G., Special relativity with acceleration, Mathematical Association of America.
The American Mathematical Monthly, Vol. 107, No. 3, 219-237, 2000.
102
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey.
2Yildiz Technical University, Department of Mathematical Engineering, Istanbul, Turkey.
Emails: [email protected], [email protected], [email protected]
Comparison of encryption and decryption algorithms through various
approaches
Murat Sari1, Meliha İpek Bulut
2, İlter Beren Kanpak
2
The aim of this paper is to produce computer codes of fundamental encryption and
decryption algorithms in a comparison way through various approaches such as Substitution,
Affine, Vigenere, Caesar and RSA. Performances of the corresponding approaches have been
deeply investigated by comparing the usage RAM and CPU times. The codes for the
encryption and decryption algorithms are written in C#. General summary of cryptography
has also been presented. Effects and performances of computer codes of encryption and
decryption algorithms for each one of the methods Affine, Vigenere, Caesar, RSA,
Substitution have been compared. For the encryption algorithms, the RSA has been seen to be
the best for performance time. For the encryption algorithms, the Affine has been seen to be
the best for CPU. For the decryption algorithms, the Substitution has been seen to be the best
for performance time. Note also that, for the decryption algorithms, the RSA has been seen to
be the best for CPU. Increasing the encryption speed by changing the encryption algorithms is
an open problem. Future work can focus on this problem.
Keywords: Cryptology, Encryption algorithms, Decryption algorithms, Substitution, Affine,
Vigenère, Caesar, RSA.
2010 AMS Classification: 14G50, 11T71
References:
1. Singh S., The Code Book, September 1999.
2. Purnama B. and Hetty R.A.H., A New Modified Caesar Cipher Cryptography Method with
Legible Ciphertext from a Message to be Encrypted, Procedia Computer Science, 59, 195-204
2015.
3. Simmons G.J., Vigenère Cipher Cryptology, Britannica, April 2017.
4. Niederreiter H., Winterhof A., Applied Number Theory, Springer, 2015
5. Kaufmann M., Computer and Information Security Handbook, 2009.
6. Paar C. and Pelzl J., Understanding Cryptography, Springer, 2010.
7. Kartalopoulos S.V., Next Generation Intelligent Optical Networks, 191 C, Springer 2008.
8. Daras N.J. (ed.), Applications of Mathematics and Informatics in Science and Engineering,
Springer Optimization and Its Applications 91, Springer, 2014.
103
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Nevşehir HBV University, Department of Mathematics, Nevşehir,TURKEY
2Ankara University, Mathematics Department, Ankara, TURKEY
E-mails: [email protected] , [email protected]
Surfaces with Constant Slope and Tubular Surfaces
Çağla RAMİS 1 and Yusuf YAYLI 2
Tubular surfaces can be characterized as a subfamily of canal surfaces with the
constant radius. In this study, we develop the endowed reduced definition of tubular surface
and give the new general parameterization by non perpendicular circle along the base curve.
Moreover, the advantage of new characterization is to yield a tubular surface without singular
points. In accordance with this purpose, we also focus to eliminate singular points by the
location of circle which moves along the base curve of surface. Mathematical description of
these surfaces enables the relation with constant slope surfaces and the creation of their
modeling on computer.
Keywords: Canal surface, tubular surface, surface with constant slope, singularity.
2010 AMS Classification: 53A05
References:
104
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 2Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey 3Balikesir University, Department of Mathematics, Balikesir, 10145, Turkey
E-mail(s): [email protected], [email protected], [email protected]
New Contributions to Fixed-Circle Results on S-Metric Spaces
Ufuk ÇELİK1, Nihal YILMAZ ÖZGÜR
2, Nihal TAŞ
3
Recently, it has been given some fixed-circle theorems on metric and S-metric spaces.
In this talk, we present some new fixed-circle theorems on S-metric spaces. We give new
examples of S-metrics and investigate some relationships between circles on metric and S-
metric spaces. Then we investigate some existence and uniqueness conditions for fixed circles
of self-mappings.
Keyword(s): Fixed circle / fixed-circle theorem / existence theorem / uniqueness theorem / S-
metric
2010 AMS Classification: Primary: 47H10, Secondary: 54H25, 55M20, 37E10.
Reference(s):
1. Gupta, A., Cyclic Contraction on S-Metric Space, International Journal of Analysis
and Applications 3, no.2, 119-130, 2013.
2. Hieu, N. T., Ly, N.T. and Dung, N.V. A Generalization of Ciric Quasi-Contractions
for Maps on S-Metric Spaces, Thai Journal of Mathematics 13, no.2, 369-380, 2015.
3. Özgür, N.Y. and Taş, N. Some fixed point theorems on S-metric spaces, Mat. Vesnik
69, no.1, 39-52, 2017.
4. Özgür, N.Y. and Taş, N. Some new contractive mappings on S-metric spaces and their
relationships with the mapping (S25), Math. Sci. 11, no.7, doi:10.1007/s40096-016-0199-4,
2017.
5. Özgür, N.Y. and Taş, N. Some fixed circle theorems on metric spaces,
arXiv:1703.00771v1 [math.MG].
6. Özgür, N.Y. and Taş, N. Some fixed circle theorems on S-metric spaces with a geometric
viewpoint, submitted for publication.
7. Ögür, N.Y. and Taş, N. Some Generalizations of Fixed Point Theorems on S-Metric
Spaces, Essays in Mathematics and Its Applications in Honor of Vladimir Arnold, New
York, Springer, 2016.
8. Sedghi, S., Shobe N. and Aliouche, A. A Generalization of Fixed Point Theorems in
S-Metric Spaces, Mat. Vesnik 64, no.3, 258-266, 2012.
105
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Kilis 7 Aralık University, Department of Mathematics, , Kilis, TURKEY 1 2 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 2 3 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 3
E-mail(s): [email protected], [email protected], [email protected]
On The Parallel Ruled Surfaces With B-Darboux Frame
Mustafa Dede1, Hatice TOZAK
2, Cumali Ekici
3
In this paper, the parallel ruled surfaces with B-Darboux frame are introduced in
Euclidean 3-space. Then some characteristic properties of the parallel ruled surfaces with B-
Darboux frame such as developability, striction point and distribution parameter are given in
E³.
Keyword(s): Parallel ruled surface/ Darboux frame/ Surfaces.
2010 AMS Classification: 53A05, 53A15,53R25
Reference(s):
1. Çöken A. C., Çiftçi Ü. and Ekici C., On parallel timelike ruled surfaces with timelike
rulings, Kuwait J. Sci. Engrg., 35(1), 21-31, 2008.
2. Gray A., Salamon S. and Abbena E., Modern differential geometry of curves and surfaces
with Mathematica, Chapman and Hall/CRC, 2006.
3. O'Neill B., Elementary differential geometry, Academic Press Inc, New York, 1996.
4. Klok F., Two moving coordinate frames for sweeping along a 3D trajectory, Comput.
Aided Geom. Des., 3(3), 217-229, 1986.
5. Darboux G., Leçons sur la theorie generale des surfaces I-II-III-IV., Gauthier-Villars, Paris,
1896.
6. Şentürk G. Y. and Yüce S., Characteristic properties of the ruled surface with Darboux
frame in E³, Kuwait J. Sci., 42(2), 14-33, 2015.
7. Hacısalihoğlu H. H., Diferensiyel geometri, İnönü Üniv. Fen Edebiyat Fak. Yayınları, 2,
1983.
8. Shin H., Yoo S. K., Cho S. K. and Chung W. H., Directional offset of a spatial curve for
practical engineering design, ICCSA, 3, 711-720, 2003.
9. Hoschek J., Integral invarianten von regelflachen, Arch. Math, XXIV, 1973.
10. Bloomenthal J., Calculation of reference frames along a space curve, Graphics Gems,
Academic Press Professional, Inc., San Diego, CA., 1990.
11. Dede M., Ekici C. and Görgülü A., Directional B-Darboux frame along a space curve.
IJARCSSE, 5, 775-780, 2015.
12. Dede M., Ekici C. and Tozak H., Directional tubular surfaces, Int. J. Algebra, 9(12), 527-
535, 2015.
13. Yüksel N., The ruled surfaces according to Bishop frame in Minkowski space, Abstr.
Appl. Anal., http://dx.doi.org/10.1155/2013/810640, 2013, 1-5, 2013.
14. Carmo P.M., Differential geometry of curves and surfaces, Prentice-Hall, Englewood
Cliffs, New York, 1976.
106
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1 Kilis 7 Aralık University, Department of Mathematics, , Kilis, TURKEY 1 2 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 2 3 Eskisehir Osmangazi University, Department of Mathematics-Computer ,Eskişehir, TURKEY 3
E-mail(s): [email protected], [email protected], [email protected]
15. Bishop R.L., There is more than one way to frame a curve, Am. Math. Mon., 82, 246-251,
1975.
16. Coquillart S., Computing offsets of B-spline curves, Comput. Aided Des., 19, 305-309,
1987.
17. Ravani T. and Ku S., Bertrand offsets of ruled surface and developable surface, Comput.
Aided Geom. Des., 23(2), 145-152, 1991.
18. Maekawa T., Patrikalakis N.M., Sakkalis T. and Yu G., Analysis and applications of pipe
surfaces, Comput. Aided Geom. Des., 15, 437-458, 1988.
19. Hlavaty V., Differentielle linien geometrie, Uitg P. Noorfhoff, Groningen, 1945.
20. Wang W., Jüttler B., Zheng D. and Liu Y., Computation of rotation minimizing frames,
ACM Trans. Graph.27, 1-18, 2008.
21. Kühnel W., Differential geometry, curves-surfaces-manifolds, Am. Math. Soc., 2002.
22. Ünlütürk Y., Çimdiker M. and Ekici C., Characteristic properties of the parallel ruled
surfaces with Darboux frame in Euclidean 3-space, Commun. Math. Model. Appl., 1(1), 26-
43, 2016.
23. Savcı Z., Görgülü A. and Ekici C., On Meusnier theorem for parallel surfaces, Commun.
Fac. Sci. Univ. Ank. S. A1 Math. Stat., 66(1),187-198, 2017.
107
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Istanbul University,Department of Mathematical Education,Istanbul, Turkey 2 Istanbul University,Department of Mathematical Education,Istanbul, Turkey
E-mail(s):[email protected], [email protected]
An ANFIS Perspective for the diagnosis of type II diabetes
Murat Kirişci1, M. Ubeydullah Saka
2
An adaptive network is a multilayer feed forward network in which each node
performs a particular function (node function) on incoming signals as well as asset of
parameters pertaining to this node. Fuzzy inference systems are the fuzzy rule based systems
which consists of a rule base, database, decision making unit, fuzzification interface and a
defuzzification interface. By embedding the fuzzy inference system into the framework of
adaptive networks, a new architecture namely Adaptive neuro fuzzy inference system
(ANFIS) is formed which combines the advantages of neural networks and fuzzy theoretic
approaches.
In this study ANFIS is presented for the diagnosis of diabetes diseases. The ANFIS
classifier is used to diagnose diabetes disease when six features defining diabetes indications
are used as inputs. The proposed ANFIS model is then evaluated and its performance is
reported. We are able to achieve significant improvement in accuracy by applying the ANFIS
model. Finally, some conclusions are drawn concerning the impacts of features on the
diagnosis of diabetes disease.
Keyword(s): diabetes, fuzzy logic, adaptive neuro-fuzzy inference system(ANFIS)
2010 AMS Classification: 68T05, 92C50, 03E72
Reference(s):
1. Zadeh L.A., Fuzzy sets, Inform. and Control, 8, 338-353, 1965.
2. Zadeh, L.A., Biological application of the theory of fuzzy sets and systems, The
proceedings of an International Symposium on Biocybernetics of the Central Nervous
System, 199-206, 1969.
3. Allahverdi, N., Design ıf Fuzzy expert systems and its applications in some medical areas,
International Journal of Applied Mathematics, Electronics and Computers, 2(1), 1-8, 2014.
4. Torres A. & Nieto J. J. Fuzzy Logic in Medicine and Bioinformatics, Journal of
Biomedicine and Biotechnology, Vol. 2006, 1–7, 2006.
5. Mahfouf M., Abbod M. F. & Linkens D. A. A survey of fuzzy logic monitoring and
control utilisation in medicine, Artificial Intelligence in Medicine, 21(1–3), 27–42, 2001.
6. Polat, K., & Gunes, S., An expert system approach based on principal component analysis
and adaptive neuro-fuzzy inference system to diagnosis of diabetes disease. Digital Signal
Processing, 17(4), 702–710, 2007.
7. Mohamed, E. I., Linderm, R., Perriello, G., Di Daniele, N., Poppl, S. J., & De Lorenzo, A.,
Predicting type 2 diabetes using an electronic nose-base artificial neural network analysis.
Diabetes Nutrition & Metabolism, 15(4),
215–221, 2002.
108
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail:[email protected]
Transitive Operator Algebras and Hyperinvariant Subspaces
Elif Demir1
In this work, I deal with the transitive and localizing operator algebras. Also I
investigate hyperinvariant subspaces.
Keyword(s): Transitive algebra, localizing algebra, hyperinvariant subspace.
2010 AMS Classification: 47L10, 47L45, 47A15
References:
1. Y. A. Abromovich, C.D. Aliprantis, An Invitation To Operator Theory, American
Mathematical Society, Rhode Island (2002).
2. C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis, A Hitchiker’s Guide, Second
Edition, Springer-Verlag, Berlin (1999).
3. J. B. Conway, A Course In Functional Analysis, Second Edition, Springer-Verlag, New
York, (1990).
4. V. I. Lomonosov, H. Radjavi, V. G. Troitsky, Sesquitransitive and Localizing Operator
Algebras, Integral Equations and Operator Theory, 60 (2008), 405-418.
5. B. P. Rynne, M. A. Youngson, Linear Functional Analysis, Springer-Verlag, London,
(2008).
6. V. G. Troitsky, Minimal vectors in arbitrary Banach spaces, Proc. American Mathematical
Society, 132(2004), 1177-1180.
109
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Anadolu University, Science Faculty, Department of Physics, Eskisehir,Turkey 2Dumlupınar University, Faculty of Art and Science, Department of Physics, Kütahya, Turkey
E-mail(s): [email protected], [email protected], [email protected]
Reformulation of Compressible Fluid Equations in Terms of Biquaternions
Süleyman Demir1, Murat Tanışlı
1, Mustafa Emre Kansu
2
In relevant literature, although Maxwell’s equations of electromagnetism have been
expressed in many mathematical forms, the same is not true for their analogous equations in
fluid mechanics. In this work, a reformulation is proposed based on biquaternions for the
Maxwell type equations of compressible fluids stimulating the biquaternionic generalization
of electric and magnetic fields in electromagnetism. After reviewing the analogy between the
structure of electrodynamics and fluid dynamics, the biquaternionic expressions of the fluid
Maxwell equations have been derived. Furthermore, the field and wave equations for fluids
have been presented in a compact and simple way.
Keywords: Biquaternion, fluid equations, Maxwell equations, field equations
2010 AMS Classification: 76A02, 76W05, 11R52
References:
1. Logan J.G, Hydrodynamic analog of the classical field equations, Phys. Fluids, 5, 868-
869, 1962.
2. Troshkin O.V, Perturbation waves in turbulent media, Comp. Maths. Math. Phys., 33,
1613-1628, 1993.
3. Marmanis H, Analogy between the Navier-Stokes equations and Maxwell's equations:
Application to turbulence, Phys. Fluids, 10, 1428-1437, 1998.
4. Kambe T, A new formulation of equations of compressible fluids by analogy with
Maxwell's equations, Fluid Dyn. Res. 42, 055502, 2010.
5. Scofield D.F., Huq P., Fluid dynamical Lorentz force law and Poynting theorem-
introduction, Fluid Dyn. Res., 46, 055513, 2014.
6. Scofield D.F., Huq P., Fluid dynamical Lorentz force law and Poynting theorem-
derivation and implications, Fluid Dyn. Res., 46, 055514, 2014.
7. Thompson R.J., Moeller T.M., A Maxwell formulation for the equations of a plasma,
Phys. Plasmas, 19, 010702, 2012.
8. Tanışlı M, Demir S., Şahin N., Octonic formulations of Maxwell type fluid equations, J.
Math. Phys., 56, 091701, 2015.
9. Demir S., Tanışlı M., Hyperbolic octonion formulation of the fluid Maxwell equations, J.
Korean Phys. Soc., 68, 616-623, 2016.
10. Demir S., Uymaz A., Tanışlı M., A new model for the reformulation of compressible
fluid equations, Chin. J. Phys., 55, 115-126, 2017.
11. Demir S., Tanışlı M., Spacetime algebra for the reformulation of fluid field equations, J.
Geo. Meth. Mod. Phys., 14, 1750075, 2017.
110
13th Algebraic Hyper structures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
2Bahçeşehir University, Department of Mathematic Engineering, İstanbul, Turkey
E-mails: [email protected], [email protected]
On The Trace Formula for a Differential Operator of Second Order with Unbounded
Operator Coefficients
Erdal GÜL
1 and Duygu ÜÇÜNCÜ
2
We investigate the spectrum of a differential operator of second order with unbounded
operator coefficients and two terms and we calculate the trace of this operator.
Keywords: Hilbert Space, Self-adjoint operator, Kernel operator, Spectrum, Essential
spectrum, Resolvent.
2010 AMS Classification: 47A10, 34L20, 34L05
References:
1. Adıgözelov E., "About the trace of the difference of two Sturm-Liouville operator with
operator coefficient," Iz. AN AZ SSR, seriya _z-tekn. i mat. nauk5, 20-24 (1976)
2. Chalilova R. Z., "On regularization of the trace of the Sturm-Liouville operator equation,"
Funks. Analiz, teoriya funksiy i ikpril.-Maha_ckala3, 154-161 (1976).
3. Maksudov . F. G., Bayramogluand M., Adıguzelov E., "On regularized trace of Sturm-
Liouville operator on a finite interval with unbounded operator coefficient," Dokl. Akad.
Nauk SSSR 30, 169-173 (1984).
4. Adıgüzelov E., Avcı H. and Gül E., "The trace formula for Sturm-Liouville operator with
Operator coefficient," JMP (Journal of Mathematical Physics) 42, No:6, 2611-2624 (2001).
5. Albayrak I., Baykal O. and Gül E., "Formula for the highly regularized trace of Sturm-
Liouville operator with unbounded operator coefficients wich has singularity", Turkish
Journal of Mathematics, Vol.25, No:2, pp.307-322 (2001)
6. Kato T., Perturbation Theory for Linear Operators (Berlin-Heidelberg-New York-Verlag,
1980).
7. Lysternikand L. A., Sobolev V. I, Elements of functional analysis, English Trans.
(NewYork: FredrickUngar, 1955).
8. Cohbergand I. C., Krein M. G., Introduction to the Theory of LinearNon-self adjoint
Operators, Translation of Mathematical Monographs, Vol. 18 (AMS, Providence, RI, 1969).
9. Maksudov F.G., Bairamoglu M. and Adigezalov E., "On asymptotics of spectrumand trace
of high order differential operator with operator coefficients, Doğa-TurkishJournal of
MathematicsVol 17, No: 2 , 113-128 (1993).
111
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Esenler, İstanbul 34220, Turkey 2Gebze Technical University, Department of Mathematics, Gebze, Kocaeli 41400, Turkey
E-mail(s): [email protected], [email protected]
Practical Stability Analyses of Nonlinear Fuzzy Dynamic Systems of Unperturbed
Systems with Initial Time Difference
Mustafa Bayram Gücen1, Coşkun Yakar
2
In this work, we have investigated the practical stability of fuzzy differential systems
of unperturbed systems and we have established a comparison result. Some practical stability
theorems is presented; in the last section, we have a comparison result in practical stability of
fuzzy differential systems of unperturbed systems via a scalar differential equation.
Keyword(s): Initial time difference, practical stability, fuzzy differential equations
2010 AMS Classification: 34D10, 34D99
Reference(s):
1. Lakshmikantham, V. and Leela, S, Differential and Integral Inequalities, Vol. I.
Academic Press, New York, 1969.
2. Lakshmikantham, V. and Leela, S, Fuzzy differential systems and the new concept of
stability. Nonlinear Dynamics and Systems Theory, 1 (2) 111-119, 2001
3. Lakshmikantham, V. and Mohapatra, R. N. Theory of Fuzzy Differential Equations.
Taylor and Francis Inc. New York, 2003.
4. Yakar, C. and Shaw, M. D. , A comparison result and Lyapunov stability criteria with
initial time difference. Dynamics of Continuous, Discrete & Impulsive Systems. Series
A, vol. 12, no. 6, 731—737, 2005.
5. Li, A., Feng, E. and Li, S., Stability and boundedness criteria for nonlinear differential
systems relative to initial time difference and applications. Nonlinear Analysis: Real
World Applications 10, 1073—1080, 2009.
112
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
2 Iskenderun Technical University, Department of Computer Engineering, Hatay, Turkey
E-mails:[email protected], [email protected]
A Numerical Scheme for Solving Nonlinear Fractional Differential Equations in The
Conformable - Derivative
Sebahat Ebru DAŞ1, Sertan ALKAN
2
Fractional Calculus is a field that involves noninteger order differential and integral
operators. The history of fractional calculus dates back to the end of the 17th century. In 1695,
half-order derivative was mentioned in a letter from L’Hopital to Leibniz [1]. Since then,
many mathematicians have contributed to the development of fractional calculus. Therefore,
many definitions for the fractional derivative are available. The most popular definitions are
Riemann-Liouville and Caputo. Additionally, recently Khalil et at. [2] introduced a new
definition of fractional derivative called the Conformable Fractional Derivative.
In our work, Sinc-Collocation Method is presented to obtain the approximate solution
of the fractional order boundary value problem with variable coefficients in the following
form
𝜇2(𝑥)𝑦′′(𝑥) + 𝜇𝛼(𝑥)𝑦(𝛼)(𝑥) + 𝜇1(𝑥)𝑦′(𝑥) + 𝜇𝛽(𝑥)𝑦(𝛽)(𝑥) + 𝜇0(𝑥)𝑦(𝑥) + 𝑛(𝑥)𝑦𝑚(𝑥) = 𝑓(𝑥)
with boundary conditions
𝑦(𝑎) = 0 , 𝑦(𝑏) = 0
where 𝑦(𝛼) and 𝑦(𝛽) are the conformable fractional derivative for 1 < 𝛼 ≤ 2 and 0 < 𝛽 ≤ 1.
Keywords: Nonlinear differential equations, conformable –derivative
2010 AMS Classification: 34A08
Reference(s):
1. Samko S.G., Kilbas A.A. , Marichev O.I., Fractional Integrals and Derivatives, Gordon
and Beach, Yverdon, 1993
2. Miller K., Ross B., An Introduction to The Fractional Calculus and Fractional Differential
Equations, New York: Wiley, 1993.
113
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mails: [email protected], [email protected]
On Locally Convex Solid Riesz Spaces
Fatma ÖZTÜRK ÇELİKER1 and Pınar ALBAYRAK
1
An ordered vector space E is called a Riesz space if every pair of vectors has a
supremum and an infimum. A locally convex (solid) topology on a vector space is a linear
topology that has a base at zero consisting of convex (solid) sets. A subset S of Riesz space E
is said to be solid if |𝑢| ≤ |𝑣| and 𝑣 ∈ 𝑆 imply 𝑢 ∈ 𝑆. A linear topology 𝜏 on a Riesz space E
that is at the same time locally solid and locally convex will be called a locally convex-solid
topology. A locally convex-solid Riesz space (𝐸, 𝜏) is a Riesz space E equipped with a locally
convex-solid topology 𝜏.
On this study we consider invariant ideals on locally convex solid Riesz spaces for
positive operators.
Keyword(s): Invariant ideals, locally convex solid Riesz Spaces
2010 AMS Classification: 47A15
References:
1. Abromovich Y.A., Aliprantis C. D., Burkinshaw O., An Invation to Operator Theory,
American Mathematical Society, 2002.
2. Abromovich Y.A., Aliprantis C. D., Burkinshaw O., The Invariant Subspace Problem:
Some Recent Advances, Rend. Istit. Mat. Univ. Trieste 29, 3-79, 1998.
3. Aliprantis C. D., Burkinshaw O., Positive Operators, Academic Press, 1985.
4. Aliprantis C. D., Burkinshaw O., Locally Solid Riesz Spaces with Applications to
Economics, American Mathematical Society, 2003.
114
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail(s):[email protected], [email protected]
On Weakly Compact-Friendly Operators
Pınar Albayrak1 and Fatma Öztürk Çeliker
1
A positive operator B:E E is said to be weakly compact-friendly if there exists a
positive operator in the commutant of B dominates a non-zero operator which in turn is
dominated by positive weakly compact operator. That is, B is weakly compact-friendly if and
only if there exists three non-zero operators R, C, K : E E with R, K positive and K
weakly compact such that
RB BR , Cx R x , and Cx K x
for each x E .
On this talk we generalized some well-known results of weakly compact-friendly
operators on Banach lattices.
Keyword(s): Invariant ideals, invariant subspaces, weakly compact-friendly operators
2010 AMS Classification: 47B65, 47A15
References:
1. Aliprantis, C.D. ,Burkinshaw, O., Positive Compact Operators on Banach Lattices, Math.
Z., 174, 289-298, 1980.
2. Aliprantis, C.D. ve Burkinshaw, O., Positive operators, Academic Pres, New
York/London,1985.
3. Gök, Ö. , Albayrak, P., On Invariant Subspaces of Weakly Compact-Friendly Operators,
International Journal of Contemporary Mathematical Sciences, Vol. 4, no:6, 259-266,2009.
115
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail:[email protected]
Hirota Type Discretization Of Clebsch Equations
Murat Turhan1
The equations of motion of a rigid body in an ideal fluid is given by the following system:
{
�̇� = 𝐱 × 𝜕𝐻
𝜕𝒑
�̇� = 𝐱 × 𝜕𝐻
𝜕𝒙+ 𝒑 ×
𝜕𝐻
𝜕𝒑
where 𝐻 ∈ 𝐶∞(ℝ6, ℝ) is a quadratic polynomial in 𝐱 and 𝒑.
Applying bilinear method and using the gauge invariance and the time reversibility of the
equations, we get gauge-invariant bilinear difference equations. Finally, we derive the explicit
discrete system by considering Hirota bilinear transformation method and present sufficient
number of the discrete conserved quantities for integrability.
Keywords: Clebsch system, discretization, bilinear form, Gröbner basis
2010 AMS Classification: 70H99
References:
1. Abraham, R., Marsden, J.E. and Ratiu, T.S., Manifolds, Tensor Analysis, and
Applications, V.75 of Applied Mathematical Sciences, Springer-Verlag. 1988.
2. http://www.asir.org
3. Lesser, M., The Analysis of Complex Nonlinear Mechanical Systems: A Computer
Algebra Assisted Approach, World Scientific, Series A, Vol.17, 1995.
4. Hirota, R., Kimura, K. and Yahagi, H., How to find the conserved quantities of
nonlinear discrete equations, J.Phys.A:Math.Gen. 34, 10377–10386, 2001.
5. Zhivkov, A., Christov, O., Effective solutions of Clebsch and C. Neumann systems,
Sitzungsberichte der Berliner Mathematischen Gesellschaft, 217–242, 2001.
6. Perelomov, A.M., A few remarks about integrability of the equations of motion of a
rigid body in ideal fluid, Phys.Lett.A 80, no:2-3, 156–158, 1980.
116
13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
2Bitlis Eren University, Department of Mathematics, Bitlis, Turkey
E-mail: [email protected] , [email protected], [email protected]
An h-deformation of the Superspace R(1|2) via a contraction
Salih Çelik 1, Sultan A. Çelik 1 and Fatma Bulut 2
A deformation of classical matrix (super) groups can be made using some facts known
from the classical. One way to obtain a deformation of classical (super) groups is to make a
deformation of (super) spaces first [1].
The one-parametric h-deformation of the algebra of coordinate functions on the
superspace 1| 2R via a contraction of the quantum superspace 1| 2qR is presented [2]. An
interesting case is that the deformation parameter h is Grassmann number.
It is well known that a matrix T in the supergroup 1| 2GL defines the linear
transformation : 1| 2 1| 2h hT R R . As a result of this we have R 1| 2hT X X .
So, the elements of the matrix T fulfill some relations. The bi-algebra structure of 1| 2hGL
is discussed.
Keywords: Quantum superspace, q-deformation, h-deformation, quantum supergroup, Hopf
superalgebra.
2010 AMS Classification: 17B37, 81R60
References:
1- Manin, Yu I., Multiparametric quantum deformation of the general linear supergroups,
Commun. Math. Phys. 123 (1989), 163-175.
2- Madore, J., An Introduction to Noncommutative Geometry and its Physical Applications,
Cambridge U. P., Cambridge, (1995).
3- Zakrzewski, S., A Hopf _-algebra of polynomials on the quantum SL(2,R) for a unitary R-
matrix, Lett. Math. Phys. 22 (1991), 287-289.
4- Ohn, C., A _-product of SL(2) and the corresponding nonstandard quantum Uh(sl(2)), Lett.
Math. Phys. 25 (1992), 85-88.
5- Kupershmidt, B.A., The quantum group GLh(2), J. Phys. A 25 (1992), L1239-1244.
6- Aghamohammadi, A., M. Khorrami and A. Shariati, h-deformation as a contraction of q-
deformation, J. Phys. A: Math. Gen. 28 (1995), L225-L231.
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
2Bitlis Eren University, Department of Mathematics, Bitlis, Turkey
E-mail: [email protected] , [email protected], [email protected]
7- Celik, S., Bicovariant differential calculus on the superspace Rq(1j2), J. Alg. and Its Appl.
15, No.9 (2016), 1650172 (17 pages)
8- Aizawa, N. and R. Chakrabarti, Noncommutative geometry of super-jordanian OSph(2/1)
covariant quantum space, J. Math. Phys. 45 (2004), 1623-1638.
9- Kac, V.: Lie Superalgebras, Adv. in Math. 26 (1977), 8-96.
10- Celik, S. and Bulut, F., A differential calculus on superspace Rh(1j2) and related topics,
Adv. Appl. Clifford Algebras 27 (2017), 1019-1030.
Salih C¸ elik, Sultan C¸ elik aa B
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13th Algebraic Hyperstructures and its Applications (AHA2017) 24-27 July, 2017, Istanbul-Turkey
Euler-Savary’s Formula on Dual Plane
Mücahit Akbıyık1, Salim Yüce2
In this work, one three Dual planes, of which two of them are moving and the other
one is fixed, are considered during the one parameter motion. In each motion; the velocities,
the relation between the velocities and the rotation poles were calculated. In addition, Euler-
Savary formula, which gives the relationship between the curvature of pole curves and
trajectory curve, were given by two different methods.
Keywords: Dual Plane, Euler Savary’s Formula, Kinematics.
2010 AMS Classification: 53A17,53A35, 53A40.
References:
1. Edmund Taylor Whittaker, A Treatise on the Analytical Dynamics of Particles and RigidBodies, Cambridge University Press. Chapter 1, 1904.
2. Joseph Stiles Beggs, Kinematics, Taylor and Francis. p. 1, 1983.3. Thomas Wallace Wright, Elements of Mechanics Including Kinematics, Kinetics and
Statics. E and FN Spon. Chapter 1, 1896. 4. A. Biewener, Animal Locomotion, Oxford University Press, 2003.5. Yaglom, I.M., A simple non-Euclidean Geometry and its Physical Basis, Springer-Verlag,
New York, 1979. 6. Yaglom I.M., Complex Numbers in Geometry, Academic Press, New York, 1968.7. Alexander J.C., Maddocks J.H., On the Maneuvering of Vehicles, SIAM J. Appl. Math.
48(1): 38-52,1988. 8. Buckley R., Whitfield E.V., The Euler-Savary Formula, The Mathematical Gazette
33(306): 297-299, 1949. 9. Dooner D.B., Griffis M.W., On the Spatial Euler-Savary Equations for Envelopes, J. Mech.
Design 129(8): 865-875, 2007. 10. Ito N., Takahashi K.,Extension of the Euler-Savary Equation to Hypoid Gears, JSME
Int.Journal. Ser C. Mech Systems 42(1): 218-224, 1999. 11. Pennock G.R., Raje N.N.,Curvature Theory for the Double Flier Eight-Bar Linkage,
Mech. Theory 39: 665-679, 2004. 12. Blaschke, W., and Müller, H.R., Ebene Kinematik, Verlag Oldenbourg, München, 1959.13. A. A. Ergin, On the one-parameter Lorentzian motion,Comm. Fac. Sci. Univ. Ankara,
Series A 40, 59--66, 1991.14. A.A.Ergin, Three Lorentzian planes moving with respect to one another and pole points,
Comm. Fac. Sci.Univ. Ankara, Series A 41,79-84,1992.1Yildiz Technical University,Department of Mathematics,Istanbul, Turkey 2Yildiz Technical University,Department of Mathematics,Istanbul, Turkey E-mails:[email protected], [email protected]
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13th Algebraic Hyperstructures and its Applications (AHA2017) 24-27 July, 2017, Istanbul-Turkey
15. I. Aytun, Euler-Savary formula for one-parameter Lorentzian plane motion and itsLorentzian geometrical interpretation, M.Sc. Thesis, Celal Bayar University, 2002.
16. T. Ikawa, Euler-Savary's formula on Minkowski geometry, Balkan Journal of Geometryand Its Applications, 8 (2), 31-36, 2003.
17. Otto Röschel, Zur Kinematik der isotropen Ebene., Journal of Geometry, 21, 146--156,1983.
18. Akar, M., Yüce, S. and Kuruoglu, N, One parameter planar motion in the Galilean plane,International Electronic Journal of Geometry, 6(1), 79-88, 2013.
19. Akbiyik, M., Yüce, S., The Moving Coordinate System And Euler-Savary's Formula ForThe One Parameter Motions On Galilean (Isotropic) Plane, International Journal ofMathematical Combinatorics(2), 88-105.
20. Kuruoğlu, N., Tutar, A., Düldül, M., On the moving coordinate system on the complexplane and pole points, Bulletin of Pure and Applied Sciences, Vol. 20 E,No1, 1-6, 2001.
21. Masal, M., Tosun, M., Pirdal, A. Z., Euler Savary formula for the one parameter motionsin the complex plane
1Yildiz Technical University,Department of Mathematics,Istanbul, Turkey 2Yildiz Technical University,Department of Mathematics,Istanbul, Turkey E-mails:[email protected], [email protected]
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
E-mail: [email protected] , [email protected]
Mistakes and Misconceptions Regarding to Natural Numbers on Secondary
Mathematics Education
Ayten Özkan1 and Erdoğan Mehmet Özkan
1
Mistakes and misconceptions regarding to natural numbers of 12th class students and
whether these mistakes and misconceptions demonstrated any significant difference
depending on the gender has been investigated in this study. This study was carried out with
60 students at 12th class who have being educated in Anatolian High Schools located in
Istanbul City Fatih Province in the education training year of 2017-2018 after completing of
the natural number subject. Cronbach Alpha coefficient reliability of the Diagnosis Test was
found as 0,90. An expert opinion was obtained for the validity. The SPSS 15 pack program
was used in order to solve the data obtained by Diagnosis Test which composed by open-
ended questions. Qualitative and quantitative researching methods were utilized in this study.
The answers given by the students were examined individually and the answers of the
students were evaluated in categories such as “correct”, “mistake”, “empty” and
“misconception”, then distribution of these students’ answers into percentage and frequency
categories were determined. Also samples within all determined mistakes and misconceptions
transferred into the computer via scanner and were submitted in findings. At the end of the
investigation, it has been determined that students had a lot of mistake and misconceptions
regarding to natural number, features belong to exponentiation, base arithmetic, prime
numbers, relative prime numbers, prime factorization of a natural number, positive dividends
of a natural number and factorial. Also these mistakes and misconceptions were determined as
not demonstrating a significant difference depending on the genders.
Keywords: Mathematics teaching, misconception, common mistakes, natural numbers
2010 AMS Classification: 97C10, 97C70
References:
1-Ercan, B. “Evaluation of the Information Related to the Concept of the Integers of the Seventh
Grade Primary School Students”, Master Thesis, Çukurova University, Social Sciences Institute,
(2010).
2-Güner, N. and Alkan, V. “Errors in Primary and Secondary Students' Answering 2010 YGS
Mathematics Questions”, Pamukkale University Journal of Education, 2, 30, (2011) ,125-140.
3- Ubuz, B., “Problems and Misconceptions of 10th and 11th Grade Students on Basic Geometry
Issues”, Hacettepe University Journal of Education, 16, 17, (1999) 95 –104.
4- Zembat İ.Ö. “What is Misconception? ", Misconceptions of Mathematical Concept and Solution
Proposals”, Edit: Özmantar,M.F. , Bingölbali, E.and Akkoç, H.,Pegem Academy:Ankara, (2010).
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13th Algebraic Hyperstructures and its Applications (AHA2017)
24-27 July, 2017, Istanbul-Turkey
1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey
2Yildiz Technical University, Marine Machinery, Istanbul,Turkey
E-mails:[email protected], vardar@ yildiz.edu.tr
On a Fuzzy Application of the Particulate Matter Estimation
Filiz Kanbay1, Nurten Vardar
2
In this study, Particulate matter PM from transit vessels passing through the
Bosphorus which connects Black sea and sea of Marmara with the length 12 sea knot are
calculated by using fuzzy inference system in MATLAB. Total particulate matters from ships
are expressed surfaces and these results allow the analysis of the data gross tone and the type
of ships.
Keywords: Particulate matter, fuzzy, surface, ship
2010 AMS Classification: 65D18, 68T27, 93B99
References:
1. Deniz C., Yalçın D. “Estimating Shipping Emissions in the Region of the Sea of Marmara,
Turkey” Science of the Total Environment 390(2008)255-261
2. Trozzi, C., Vaccoro,R., 1998. Methodologies for Estimating Air Pollutant Emissions From
Ships. Techne Report MEET. (Methodologies for Estimating Air Pollutant Emissions from
Transport)RF98
3. Kesgin, U., Vardar, N., 2001. A study on Exhaust gas Emissions from Ships in Turkish
Straits Atmospheric Environment 35, pp. 1863-1870.
4. Kılıç, A., 2009. Marmara Denizi’nde Gemilerden Kaynaklanan Egzoz Emisyonları, BAÜ
FBE Dergisis Cilt:11,Sayı:2, pp.124-134
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