Research Article The Lattice of Intuitionistic Fuzzy Filters in Residuated Lattices Zhen Ming Ma School of Science, Linyi University, Linyi, Shandong 276005, China Correspondence should be addressed to Zhen Ming Ma; [email protected] Received 6 November 2013; Accepted 10 April 2014; Published 29 April 2014 Academic Editor: Ch. Tsitouras Copyright © 2014 Zhen Ming Ma. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e notion of tip-extended pair of intuitionistic fuzzy filters is introduced by which it is proved that the set of all intuitionistic fuzzy filters in a residuated lattice forms a bounded distributive lattice. 1. Introduction Nowadays, it is generally accepted that in fuzzy logic the algebraic structure should be a residuated lattice which was introduced by Ward and Dilworth [1]. Some other logical algebras such as MTL-algebras [2], BL-algebras [3], MV- algebras [4], G-algebras, Π-algebras, and NM-algebras [2], which are also called 0 -algebras [5], are all able to be con- sidered particular classes of residuated lattices. (For details, see, e.g., [6].) Filters are an important tool to study these logical algebras and the completeness of the corresponding nonclassical log- ics. On the one hand, filters are closely related to congruence relations with which one can associate quotient algebras [7]; on the other hand, various filters correspond to various sets of provable formula [3, 4]. A filter is also called a deductive system in BL-algebras [8]. It has been widely investigated in residuated lattices [7, 9–11] and particular residuated lattices [2, 3, 6, 8, 12–15]. Since Rosenfeld [16] applied the notion of fuzzy sets [17] to abstract algebra and introduced the notion of fuzzy subgroups, the literature of various fuzzy algebraic concepts has been growing very rapidly [18]. In particular, in [19–21], the notion of tip-extended pair of fuzzy sets was introduced to investigate the lattices of all fuzzy normal subgroups and -ideals. e notion of fuzzy filters was introduced, and some properties of them were obtained [22–24]. Moreover, based on the notion of intuitionistic fuzzy sets (IFS) proposed by Atanassov [25], the concept of the intuitionistic fuzzy filter in BL-algebras was introduced in [26]. However, the study of residuated lattices from the point of lattice theory is less frequent. In this paper, the intuitionistic fuzzy filter theory in residuated lattices is developed. is paper is organized as follows: in Section 2, some basic concepts and properties of intuitionistic fuzzy sets and intuitionistic fuzzy filters in residuated lattices are recalled. In Section 3, by introducing the notion of tip-extended pair of intuitionistic fuzzy filters, it is proved that the set of all intuitionistic fuzzy filters forms a bounded distributive lattice. e last section concludes this paper. 2. Preliminaries e concepts of residuated lattices and intuitionistic fuzzy filters will be extensively used in the sequel. erefore, we recall their definitions and summarize their main properties. Let ̸ =0. A mapping : → [0, 1] is called a fuzzy set [17]. Let and be fuzzy sets on . en tip-extended pair of and [19, 20] can be defined by () = { () , ̸ = 1, (1) ∨ (1) , = 1, () = { () , ̸ = 1, (1) ∨ (1) , = 1. (1) Let , V : → [0,1] be two fuzzy sets satisfying 0≤ () + V () ≤ 1 for all ∈. en = ( , V ) is called an intuitionistic fuzzy set [25] (or equivalently denoted by Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 798670, 6 pages http://dx.doi.org/10.1155/2014/798670
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Research ArticleThe Lattice of Intuitionistic Fuzzy Filters in Residuated Lattices
Zhen Ming Ma
School of Science Linyi University Linyi Shandong 276005 China
Correspondence should be addressed to Zhen Ming Ma dmgywto126com
Received 6 November 2013 Accepted 10 April 2014 Published 29 April 2014
Academic Editor Ch Tsitouras
Copyright copy 2014 Zhen Ming Ma This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The notion of tip-extended pair of intuitionistic fuzzy filters is introduced by which it is proved that the set of all intuitionistic fuzzyfilters in a residuated lattice forms a bounded distributive lattice
1 Introduction
Nowadays it is generally accepted that in fuzzy logic thealgebraic structure should be a residuated lattice which wasintroduced by Ward and Dilworth [1] Some other logicalalgebras such as MTL-algebras [2] BL-algebras [3] MV-algebras [4] G-algebras Π-algebras and NM-algebras [2]which are also called 119877
0-algebras [5] are all able to be con-
sidered particular classes of residuated lattices (For detailssee eg [6])
Filters are an important tool to study these logical algebrasand the completeness of the corresponding nonclassical log-ics On the one hand filters are closely related to congruencerelations with which one can associate quotient algebras [7]on the other hand various filters correspond to various setsof provable formula [3 4] A filter is also called a deductivesystem in BL-algebras [8] It has been widely investigated inresiduated lattices [7 9ndash11] and particular residuated lattices[2 3 6 8 12ndash15]
Since Rosenfeld [16] applied the notion of fuzzy sets[17] to abstract algebra and introduced the notion of fuzzysubgroups the literature of various fuzzy algebraic conceptshas been growing very rapidly [18] In particular in [19ndash21]the notion of tip-extended pair of fuzzy sets was introducedto investigate the lattices of all fuzzy normal subgroups and119871-ideals
The notion of fuzzy filters was introduced and someproperties of them were obtained [22ndash24] Moreover basedon the notion of intuitionistic fuzzy sets (IFS) proposed byAtanassov [25] the concept of the intuitionistic fuzzy filterin BL-algebras was introduced in [26] However the study
of residuated lattices from the point of lattice theory is lessfrequent
In this paper the intuitionistic fuzzy filter theory inresiduated lattices is developed This paper is organized asfollows in Section 2 some basic concepts and propertiesof intuitionistic fuzzy sets and intuitionistic fuzzy filters inresiduated lattices are recalled In Section 3 by introducingthe notion of tip-extended pair of intuitionistic fuzzy filtersit is proved that the set of all intuitionistic fuzzy filters formsa bounded distributive lattice The last section concludes thispaper
2 Preliminaries
The concepts of residuated lattices and intuitionistic fuzzyfilters will be extensively used in the sequel Therefore werecall their definitions and summarize their main properties
Let 119880 = 0 A mapping 119891 119880 rarr [0 1] is called a fuzzy set[17] Let 119891 and 119892 be fuzzy sets on 119880 Then tip-extended pairof 119891 and 119892 [19 20] can be defined by
119891119892(119909) =
119891 (119909) 119909 = 1
119891 (1) or 119892 (1) 119909 = 1
119892119891(119909) =
119892 (119909) 119909 = 1
119892 (1) or 119891 (1) 119909 = 1
(1)
Let 119906119860 V119860 119880 rarr [0 1] be two fuzzy sets satisfying 0 le
119906119860(119909) + V
119860(119909) le 1 for all 119909 isin 119880 Then 119860 = (119906
119860 V119860) is called
an intuitionistic fuzzy set [25] (or equivalently denoted by
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 798670 6 pageshttpdxdoiorg1011552014798670
2 Journal of Applied Mathematics
119860 = ⟨119909 119906119860(119909) V
119860(119909)⟩ | 119909 isin 119880) The family of all
intuitionistic fuzzy sets on 119880 will be denoted by IFS(119880)Basic operations on intuitionistic fuzzy sets are defined in
the following wayLet 119860 119861 isin IFS(119880) One has
119860 cap 119861 = (119906119860and 119906
119861 V119860or V
119861)
119860 cup 119861 = (119906119860or 119906
119861 V119860and V
119861)
119860 sube 119861 iff 119906119860le 119906
119861 V
119860ge V
119861
119860 supe 119861 iff 119860 sube 119861
119860 = 119861 iff 119860 supe 119861 119860 sube 119861
(2)
Definition 1 (see [3]) A residuated lattice is an algebra119871 = (119871 and or otimes rarr 0 1) such that (119871 and or 0 1) is a boundedlattice with the least element 0 and the greatest element 1(119871 otimes 1) is a commutative monoid and (otimes rarr ) forms anadjoint pair that is 119909 otimes 119910 le 119911 if and only if 119909 le 119910 rarr 119911
for all 119909 119910 119911 isin 119871
A nonempty subset119865 of119871 is called a filter of119871 if (i)forall119909 119910 isin119865 119909 otimes 119910 isin 119865 (ii) forall119909 isin 119865 119910 isin 119871 119909 le 119910 implies 119910 isin 119865 orequivalently (iii) 1 isin 119865 and (iv) forall119909 isin 119865 119910 isin 119871 119909 rarr 119910 isin 119865
implies 119910 isin 119865The following alternative definitions of intuitionistic
fuzzy filters were proved in [26] but they can be similarlyverified in residuated lattices
Definition 2 Let119860 isin IFS(119871)Then119860 is called an intuitionisticfuzzy filter if
(1) 119906119860(119909) le 119906
119860(1) V
119860(119909) ge V
119860(1) for all 119909 isin 119871
(2) 119906119860(119909) and 119906
119860(119909 rarr 119910) le 119906
119860(119910) for all 119909 119910 isin 119871
(3) V119860(119909) or V
119860(119909 rarr 119910) ge V
119860(119910) for all 119909 119910 isin 119871
The set of all intuitionistic fuzzy filters on a residuatedlattice 119871 will be denoted by IFF(119871)
Theorem 3 Let 119860 isin 119868119865119878(119871) Then 119860 is an intuitionistic fuzzyfilter if and only if 119909 otimes 119910 le 119911 implies 119906
119860(119909) and 119906
119860(119910) le 119906
119860(119911)
and V119860(119909) or V
119860(119910) ge V
119860(119911) for all 119909 119910 119911 isin 119871
Theorem 4 Let 119860 isin 119868119865119878(119871) Then 119860 is an intuitionistic fuzzyfilter if and only if the following assertions hold
(1) 119909 le 119910 implies 119906119860(119909) le 119906
119860(119910) and V
119860(119909) ge V
119860(119910) for
all 119909 119910 isin 119871(2) 119906
119860(119909)and119906
119860(119910) le 119906
119860(119909otimes119910) and V
119860(119909)orV
119860(119910) ge V
119860(119909otimes
119910) for all 119909 119910 isin 119871
3 Lattice of Intuitionistic Fuzzy Filters
In this section we mainly investigate the lattice of allintuitionistic fuzzy filters by introducing the notion of tip-extended pair of intuitionistic fuzzy sets
The following lemma is obvious but necessary
Lemma 5 Let 119860 119861 be intuitionistic fuzzy filters of 119871 Then sois 119860 cap 119861
For 119860 isin IFS(119871) the intersection of all intuitionistic fuzzyfilters containing119860 is called the generated intuitionistic fuzzyfilter by 119860 denoted as ⟨119860⟩
Theorem 6 Let 119860 isin 119868119865119878(119871) Define a new intuitionistic fuzzyset 119861 by 119861 = (119906
119861 V119861) where
119906119861 (119909) = ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
V119861(119909) = ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
V119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
(3)
for all 119909 isin 119871 Then 119861 = ⟨119860⟩
Proof We complete the proof by two steps Firstly we verifythat 119861 is an intuitionistic fuzzy filter For all 119909 119910 isin 119871 suchthat 119909 le 119910 the definition of 119861 yields that 119906
119861(119909) le 119906
119861(119910) and
V119861(119909) ge V
119861(119910) For all 119909 119910 isin 119871 we have
119906119861(119909) and 119906
119861(119910)
= ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
and ⋁
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198871) and sdot sdot sdot and 119906
119860(119887119898)
= ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
⋁
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
and 119906119860(1198871) and sdot sdot sdot and 119906
119860(119887119898)
le ⋁
1198881otimessdotsdotsdototimes119888119899le119909otimes119910
119888119894isin119871119896isin119873
119906119860(1198881) and sdot sdot sdot and 119906
119860(119888119896)
= 119906119861(119909 otimes 119910)
V119861(119909) or V
119861(119910)
= ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
or ⋀
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198871) or sdot sdot sdot or 119906
119860(119887119898)
= ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
⋀
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
or 119906119860(1198871) or sdot sdot sdot or 119906
119860(119887119898)
ge ⋀
1198881otimessdotsdotsdototimes119888119899le119909otimes119910
119888119894isin119871119896isin119873
119906119860(1198881) or sdot sdot sdot or 119906
119860(119888119896)
= 119906119861(119909 otimes 119910)
(4)
Thus 119861 is an intuitionistic fuzzy filter
Journal of Applied Mathematics 3
Secondly let 119862 be an intuitionistic fuzzy filter suchthat 119862 supe 119860 By the definition of intuitionistic fuzzy filter itholds that
119906119861(119909) = ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
le ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861) and sdot sdot sdot and 119906
119862(119886119899)
le ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861otimes sdot sdot sdot otimes 119886
119899)
le 119906119862(119909)
V119861 (119909) = ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
ge ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861) or sdot sdot sdot or 119906
119862(119886119899)
ge ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861otimes sdot sdot sdot otimes 119886
119899)
ge 119906119862 (119909)
(5)
and hence 119861 sube 119862 Thus 119861 = ⟨119860⟩
Example 7 Let 119871 = 0 119886 119887 1 with 0 lt 119886 lt 119887 lt 1 Theoperations otimes and rarr are defined as
otimes 0 119886 119887 1
0 0 0 0 0
119886 0 0 119886 119886
119887 0 119886 119887 119887
1 0 119886 119887 1
rarr 0 119886 119887 1
0 1 1 1 1
119886 119886 1 1 1
119887 0 119886 1 1
1 0 119886 119887 1
(6)
Define 119906119860 V119860
119871 rarr [0 1] as 119906119860(0) = 02 V
119860(0) =
07 119906119860(119886) = 05 V
119860(119886) = 045 119906
119860(119887) = 06 V
119860(119887) =
04 119906119860(1) = 08 and V
119860(1) = 015 Since 05 = 119906
119860(119886) and
119906119860(119886) lt 119906
119860(1198862) = 02 119891 is not a fuzzy filter It is routine
to verify that ⟨119860⟩ is an intuitionistic fuzzy filter where119906⟨119860⟩(0) = 119906
⟨119860⟩(119886) = 05 V
⟨119860⟩(0) = V
⟨119860⟩(119886) = 04 119906
⟨119860⟩(119887) =
06 V⟨119860⟩(119887) = 03 119906
⟨119860⟩(1) = 08 and V
⟨119860⟩(1) = 015 from the
above theorem
Lemma 8 Let 119886 119887 119904 119905 isin [0 1] such that 0 le 119886 + 119887 le 1 and0 le 119904 + 119905 le 1 Then 0 le 119886 or 119904 + 119887 and 119905 le 1
Proof Not losing the generality we assume that 119886 le 119904 Then119886 or 119904 + 119887 and 119905 le 119904 + 119905 le 1 It is obvious that 0 le 119886 or 119904 + 119887 and 119905Thus it holds that 0 le 119886 or 119904 + 119887 and 119905 le 1
Theorem 9 Let 119860 be an intuitionistic fuzzy filter of 119871 and forall 119905 119904 isin [0 1] such that 0 le 119905 + 119904 le 1 Then 119860119905119904
= (119906119905
119860 V119904119860) is
an intuitionistic fuzzy filter where
119906119905
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119905 119909 = 1
V119904119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and 119904 119909 = 1
(7)
Proof It follows from Lemma 8 that 119860119905119904isin IFS(119871) Now we
prove that 119860119905119904 is an intuitionistic fuzzy filterIf 119909 le 119910 we consider the following two cases
Case 1 (119910 = 1) It is obvious that 119906119905119860(119909) le 119906
119905
119860(1) = 119906
119905
119860(119910)
V119905119860(119909) ge V119905
119860(1) = V119905
119860(119910)
Case 2 (119910 = 1) The definition of 119860119905119904 leads that 119906119905119860(119909) =
119906119860(119909) le 119906
119860(119910) = 119906
119905
119860(119910) V119905
119860(119909) = V
119860(119909) ge V
119860(119910) = V119905
119860(119910)
Thus 119906119905119860(119909) le 119906
119905
119860(119910) V119905
119860(119909) ge V119905
119860(119910)
Let 119909 119910 isin 119871 We consider the following two cases
Case 1 (119909 otimes 119910 = 1) If 119909 = 119910 = 1 it is obvious that 119906119905119860(119909) and
119906119905
119860(119910) le 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) ge V119905
119860(119909 otimes 119910)
If 119909 = 1 119910 = 1 or 119909 = 1 119910 = 1 it is a contradictionIf 119909 = 1 119910 = 1 it holds that 119906119905
119860(119909) and 119906
119905
119860(119910) = 119906
119860(119909) and
119906119860(119910) le 119906
119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or
119910119860(119909) ge V
119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
Case 2 (119909 otimes 119910 = 1) If 119909 = 119910 = 1 it is a contradictionIf 119909 = 1 119910 = 1 or 119909 = 1 119910 = 1 it is obvious that 119906119905
119860(119909) and
119906119905
119860(119910) le 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) ge V119905
119860(119909 otimes 119910)
If 119909 = 1 119910 = 1we have 119906119905119860(119909) and 119906
119905
119860(119910) = 119906
119860(119909) and 119906
119860(119910) le
119906119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or 119910
119860(119909) ge
V119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
All in all it yields that 119906119905119860(119909)and119906
119905
119860(119910) = 119906
119860(119909)and119906
119860(119910) le
119906119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or 119910
119860(119909) ge
V119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
Thus 119860119905119904 is an intuitionistic fuzzy filter
For given 119860 119861 isin IFS(119871) the operation ⋆ is defined by
119860⋆119861 = (119906119860otimes119906
119861 V119860⊙V
119861) (8)
where
119906119860otimes119906
119861(119909) = ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
V119860⊙V
119861(119909) = ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
(9)
Furthermore the tip-extended pair for intuitionisticfuzzy sets 119860 and 119861 are defined by
119860119861= (119906
119906119861
119860 VV119861119860) (10)
4 Journal of Applied Mathematics
where
119906119906119861
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119906119861 (1) 119909 = 1
VV119861119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and V
119861(1) 119909 = 1
119861119860= (119906
119906119860
119861 VV119860119861)
(11)
where
119906119906119860
119861(119909) =
119906119861(119909) 119909 = 1
119906119861 (1) or 119906119860 (1) 119909 = 1
VV119860119861(119909) =
V119861 (119909) 119909 = 1
V119861(1) and V
119860(1) 119909 = 1
(12)
Theorem 10 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860isin 119868119865119865(119871)
Proof It is obvious that 119906119906119861119860otimes119906
119906119860
119861is order-preserving and
VV119861119860⊙VV119860
119861is order-reserving For all 119909 119910 isin 119871 we have
119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910)
= ⋁
119906otimesVle119909otimes119910119906
119906119861
119860(119906) and 119906
119906119860
119861(V)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901 otimes 119903) and 119906
119906119860
119861(119902 otimes 119904)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901) and 119906
119906119861
119860(119903) and 119906
119906119860
119861(119902) and 119906
119906119860
119861(119904)
= ⋁
119901otimes119902le119909
119906119906119861
119860(119901) and 119906
119906119860
119861(119902) and ⋁
119903otimes119904le119910
119906119906119861
119860(119903) and 119906
119906119860
119861(119904)
= 119906119906119861
119860otimes119906
119906119860
119861(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119910)
VV119861119860⊙VV119860
119861(119909 otimes 119910)
= ⋀
119906otimesVle119909otimes119910VV119861119860(119906) or VV119860
119861(V)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901 otimes 119903) or VV119860
119861(119902 otimes 119904)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901) or VV119861
119860(119903) or VV119860
119861(119902) or VV119860
119861(119904)
= ⋀
119901otimes119902le119909
VV119861119860(119901) or VV119860
119861(119902) or ⋀
119903otimes119904le119910
VV119861119860(119903) or VV119860
119861(119904)
= VV119861119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
(13)
and hence 119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910) ge 119906
119906119861
119860otimes119906
119906119860
119861(119909) and
119906119906119861
119860otimes119906
119906119860
119861(119910) VV119861
119860⊙VV119860
119861(119909 otimes 119910) le VV119861
119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
Thus 119860119861⋆119861
119860isin IFF(119871)
Theorem 11 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Proof It is easy to prove that 119860 119861 sube 119860119861⋆119861
119860 and hence 119860 cup
119861 sube 119860119861⋆119861
119860 Thus ⟨119860 cup 119861⟩ sube 119860119861⋆119861
119860Assume that 119862 isin IFF(119871) such that 119860 cup 119861 sube 119862 If 119909 = 1
we have 119906119906119861119860otimes119906
119906119860
119861(1) = 119906
119860(1) or 119906
119861(1) le 119906
119862(1) VV119861
119860⊙VV119860
119861(1) =
V119860(1) and V
119861(1) ge V
119862(1) If 119909 lt 1 it holds that
119906119906119861
119860otimes119906
119906119860
119861(119909)
= ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119906119861
119860(119910) and 119906
119906119860
119861(119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119860(119910) and 119906
119861 (119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861 (119911)
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119910) and 119906
119862(119911) or ⋁
119910le119909
119906119862(119910) or ⋁
119911le119909
119906119862(119911)
= ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
le 119906119862(119909)
VV119861119860⊙VV119860
119861(119909)
= ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
VV119861119860(119910) or VV119860
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119860(119910) or V
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119910) or V
119862 (119911) and ⋀
119910le119909
V119862(119910) and ⋀
119911le119909
V119862 (119911)
= ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
ge V119862 (119909)
(14)
It follows fromTheorem 10 that 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Associating with the above results we prove the maintheorem here For 119860 119861 isin IFF(119871) the operations ⊓ and ⊔ onIFF(119871) are defined by
119860 ⊓ 119861 = 119860 cap 119861 119860 ⊔ 119861 = 119860119861⋆119861
119860 (15)
Theorem 12 (IFF(119871) ⊓ ⊔ 0 119871) is a bounded distributivelattice
Proof We only verify the distributivity Obviously it holdsthat 119862 ⊓ (119860 ⊔ 119861) supe (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) so we only prove
Journal of Applied Mathematics 5
that 119862 ⊓ (119860 ⊔ 119861) sube (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) Assume that 119909 isin 119871
for 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861(119909) le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860(119909) we
consider the following two cases
Case 1 (119909 = 1) We have
119906119862and 119906
119860⊔119861(1) = 119906
119862(1) and 119906
119906119861
119860otimes119906
119906119860
119861(1)
= 119906119862 (1) and (119906119860 (1) or 119906119861 (1))
= (119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(1)
V119862or V
119860⊔119861(1) = V
119862(1) or VV119861
119860⊙VV119860
119861(1)
= V119862 (1) or (V119860 (1) and V
119861 (1))
= (V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(1)
(16)
Case 2 (119909 = 1) It holds that
119906119862(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119909)
= 119906119862 (119909) and ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119906119862(119909) and 119906
119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862 (119909) and 119906
119906119861
119860(1) and 119906119861 (119909)
or 119906119862(119909) and 119906
119860(119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862 (119909) and 119906119860 (119910) and 119906119862 (119909) and 119906119861 (119911)
or 119906119862(119909) and 119906
119862(1) and 119906
119906119861
119860(1) and 119906
119861(119909)
or 119906119862 (119909) and 119906119862 (1) and 119906119860 (119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862(1) and (119906
119860(1) or 119906
119861(1))
and [(119906119862 (119909) and 119906119861 (119909)) or (119906119862 (119909) and 119906119860 (119909))]
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or [(119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))]
and [(119906119862(119909) and 119906
119861(119909)) or (119906
119862(119909) and 119906
119860(119909))]
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
or [(119906119862and 119906
119860)119906119862and119906119861
(1) and (119906119862and 119906
119860) (119910 or 119909)]
or [(119906119862and 119906
119861)119906119862and119906119860
(1) and (119906119862and 119906
119861) (119911 or 119909)]
= ⋁
119910otimes119911le119909
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
(17)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋁
1199101015840otimes1199111015840le119909
(119906119862and 119906
119860)119906119862and119906119861
(1199101015840) and (119906
119862and 119906
119861)119906119862and119906119860
(1199111015840)
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(119909)
V119862(119909) or VV119861
119860⊙VV119860
119861(119909)
= V119862 (119909) or ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
V119862(119909) or VV119861
119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) and V
119860(119909) and VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or V
119862(1) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) or V
119862(1) or V
119860(119909) or VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862 (119909) or V
119860(119910) or V
119862 (119909) or V119861 (119911)
and V119862 (1) or (V119860 (1) and V
119861 (1))
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and [(V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))]
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
6 Journal of Applied Mathematics
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
and [(V119862or V
119860)V119862orV119861
(1) or (V119862or V
119860) (119910 or 119909)]
and [(V119862or V
119861)V119862orV119860
(1) and (V119862or V
119861) (119911 or 119909)]
= ⋀
119910otimes119911le119909
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
(18)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋀
1199101015840otimes1199111015840le119909
(V119862or V
119860)V119862orV119861
(1199101015840) or (V
119862or V
119861)V119862orV119860
(1199111015840)
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(119909)
(19)
Thus 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860 and
V119862(119909) or VV119861
119860⊙VV119860
119861(119909) ge (V
119862or V
119860)V119862orV119861 ⊙(V
119862or V
119861)V119862orV119860 that is
the distributivity holds
4 Conclusions
In this paper by the notion of tip-extended pair of intuitionis-tic fuzzy sets it is verified that the set of all intuitionistic fuzzyfilters forms a bounded distributive lattice
Future research will focus on the lattice of fuzzy notionsin other algebras by tip-extended pair
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by AMEP of Linyi Universityand the Natural Science Foundation of Shandong Province(Grant no ZR2013FL006)
References
[1] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[2] F Esteva and L Godo ldquoMonoidal 119905-norm based logic towards alogic for left-continuous t-normsrdquo Fuzzy Sets and Systems vol124 no 3 pp 271ndash288 2001
[3] P Hajek Metamathematics of Fuzzy Logic Kluwer AcademicDordrecht The Netherlands 1998
[4] C C Chang ldquoAlgebraic analysis of many valued logicsrdquo Trans-actions of the American Mathematical Society vol 88 pp 467ndash490 1958
[5] G J Wang An Introduction to Mathematical Logic and Resolu-tion Principle Science in China Press Beijing China 2003
[6] E Turunen Mathematics Behind Fuzzy Logic Physica Heidel-berg Germany 1999
[7] B van Gasse G Deschrijver C Cornelis and E E KerreldquoFilters of residuated lattices and triangle algebrasrdquo InformationSciences vol 180 no 16 pp 3006ndash3020 2010
[8] E Turunen ldquoBoolean deductive systems of BL-algebrasrdquoArchive forMathematical Logic vol 40 no 6 pp 467ndash473 2001
[9] M Kondo ldquoFilters on commutative residuated latticesrdquo inIntegrated Uncertainty Management and Applications vol 68pp 343ndash347 Springer Berlin Germany 2010
[10] L Liu and K Li ldquoBoolean filters and positive implicative filtersof residuated latticesrdquo Information Sciences vol 177 no 24 pp5725ndash5738 2007
[11] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquoInformation Sciences vol 180 no 19 pp 3614ndash3632 2010
[12] MHaveshki A B Saeid and E Eslami ldquoSome types of filters in119861119871 algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006
[13] M Kondo andW A Dudek ldquoFilter theory of BL algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008
[14] A B Saeid and S Motamed ldquoNormal filters in BL-algebrasrdquoWorld Applied Sciences Journal vol 7 pp 70ndash76 2009
[15] E Turunen ldquoBL-algebras of basic fuzzy logicrdquoMathware amp SoftComputing vol 6 no 1 pp 49ndash61 1999
[16] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 no 3 pp 512ndash517 1971
[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
[18] J N Mordeson and D S Malik Fuzzy Commutative AlgebraWorld Scientific London UK 1998
[19] T Head ldquoA metatheorem for deriving fuzzy theorems fromcrisp versionsrdquo Fuzzy Sets and Systems vol 73 no 3 pp 349ndash358 1995
[20] T Head ldquoErratum to ldquoA metatheorem for deriving fuzzytheorems from crisp versionsrsquordquo Fuzzy Sets and Systems vol 79no 2 pp 277ndash278 1996
[21] I Jahan ldquoThe lattice of 119871-ideals of a ring is modularrdquo Fuzzy Setsand Systems vol 199 pp 121ndash129 2012
[22] Y B Jun Y Xu and X H Zhang ldquoFuzzy filters of MTL-algebrasrdquo Information Sciences vol 175 no 1-2 pp 120ndash1382005
[23] L Liu and K Li ldquoFuzzy filters of 119861119871-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005
[24] J L Zhang and H J Zhou ldquoFuzzy filters on the residuatedlatticesrdquo New Mathematics and Natural Compution vol 2 no1 pp 11ndash28 2006
[25] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[26] Z Xue Y XiaoW Liu H Cheng and Y Li ldquoIntuitionistic fuzzyfilter theory of 119861119871-algebrasrdquo International Journal of MachineLearning and Cybernetics vol 4 no 6 pp 659ndash669 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
119860 = ⟨119909 119906119860(119909) V
119860(119909)⟩ | 119909 isin 119880) The family of all
intuitionistic fuzzy sets on 119880 will be denoted by IFS(119880)Basic operations on intuitionistic fuzzy sets are defined in
the following wayLet 119860 119861 isin IFS(119880) One has
119860 cap 119861 = (119906119860and 119906
119861 V119860or V
119861)
119860 cup 119861 = (119906119860or 119906
119861 V119860and V
119861)
119860 sube 119861 iff 119906119860le 119906
119861 V
119860ge V
119861
119860 supe 119861 iff 119860 sube 119861
119860 = 119861 iff 119860 supe 119861 119860 sube 119861
(2)
Definition 1 (see [3]) A residuated lattice is an algebra119871 = (119871 and or otimes rarr 0 1) such that (119871 and or 0 1) is a boundedlattice with the least element 0 and the greatest element 1(119871 otimes 1) is a commutative monoid and (otimes rarr ) forms anadjoint pair that is 119909 otimes 119910 le 119911 if and only if 119909 le 119910 rarr 119911
for all 119909 119910 119911 isin 119871
A nonempty subset119865 of119871 is called a filter of119871 if (i)forall119909 119910 isin119865 119909 otimes 119910 isin 119865 (ii) forall119909 isin 119865 119910 isin 119871 119909 le 119910 implies 119910 isin 119865 orequivalently (iii) 1 isin 119865 and (iv) forall119909 isin 119865 119910 isin 119871 119909 rarr 119910 isin 119865
implies 119910 isin 119865The following alternative definitions of intuitionistic
fuzzy filters were proved in [26] but they can be similarlyverified in residuated lattices
Definition 2 Let119860 isin IFS(119871)Then119860 is called an intuitionisticfuzzy filter if
(1) 119906119860(119909) le 119906
119860(1) V
119860(119909) ge V
119860(1) for all 119909 isin 119871
(2) 119906119860(119909) and 119906
119860(119909 rarr 119910) le 119906
119860(119910) for all 119909 119910 isin 119871
(3) V119860(119909) or V
119860(119909 rarr 119910) ge V
119860(119910) for all 119909 119910 isin 119871
The set of all intuitionistic fuzzy filters on a residuatedlattice 119871 will be denoted by IFF(119871)
Theorem 3 Let 119860 isin 119868119865119878(119871) Then 119860 is an intuitionistic fuzzyfilter if and only if 119909 otimes 119910 le 119911 implies 119906
119860(119909) and 119906
119860(119910) le 119906
119860(119911)
and V119860(119909) or V
119860(119910) ge V
119860(119911) for all 119909 119910 119911 isin 119871
Theorem 4 Let 119860 isin 119868119865119878(119871) Then 119860 is an intuitionistic fuzzyfilter if and only if the following assertions hold
(1) 119909 le 119910 implies 119906119860(119909) le 119906
119860(119910) and V
119860(119909) ge V
119860(119910) for
all 119909 119910 isin 119871(2) 119906
119860(119909)and119906
119860(119910) le 119906
119860(119909otimes119910) and V
119860(119909)orV
119860(119910) ge V
119860(119909otimes
119910) for all 119909 119910 isin 119871
3 Lattice of Intuitionistic Fuzzy Filters
In this section we mainly investigate the lattice of allintuitionistic fuzzy filters by introducing the notion of tip-extended pair of intuitionistic fuzzy sets
The following lemma is obvious but necessary
Lemma 5 Let 119860 119861 be intuitionistic fuzzy filters of 119871 Then sois 119860 cap 119861
For 119860 isin IFS(119871) the intersection of all intuitionistic fuzzyfilters containing119860 is called the generated intuitionistic fuzzyfilter by 119860 denoted as ⟨119860⟩
Theorem 6 Let 119860 isin 119868119865119878(119871) Define a new intuitionistic fuzzyset 119861 by 119861 = (119906
119861 V119861) where
119906119861 (119909) = ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
V119861(119909) = ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
V119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
(3)
for all 119909 isin 119871 Then 119861 = ⟨119860⟩
Proof We complete the proof by two steps Firstly we verifythat 119861 is an intuitionistic fuzzy filter For all 119909 119910 isin 119871 suchthat 119909 le 119910 the definition of 119861 yields that 119906
119861(119909) le 119906
119861(119910) and
V119861(119909) ge V
119861(119910) For all 119909 119910 isin 119871 we have
119906119861(119909) and 119906
119861(119910)
= ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
and ⋁
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198871) and sdot sdot sdot and 119906
119860(119887119898)
= ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
⋁
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
and 119906119860(1198871) and sdot sdot sdot and 119906
119860(119887119898)
le ⋁
1198881otimessdotsdotsdototimes119888119899le119909otimes119910
119888119894isin119871119896isin119873
119906119860(1198881) and sdot sdot sdot and 119906
119860(119888119896)
= 119906119861(119909 otimes 119910)
V119861(119909) or V
119861(119910)
= ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
or ⋀
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198871) or sdot sdot sdot or 119906
119860(119887119898)
= ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
⋀
1198871otimessdotsdotsdototimes119887
119898le119910
119887119894isin119871119898isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
or 119906119860(1198871) or sdot sdot sdot or 119906
119860(119887119898)
ge ⋀
1198881otimessdotsdotsdototimes119888119899le119909otimes119910
119888119894isin119871119896isin119873
119906119860(1198881) or sdot sdot sdot or 119906
119860(119888119896)
= 119906119861(119909 otimes 119910)
(4)
Thus 119861 is an intuitionistic fuzzy filter
Journal of Applied Mathematics 3
Secondly let 119862 be an intuitionistic fuzzy filter suchthat 119862 supe 119860 By the definition of intuitionistic fuzzy filter itholds that
119906119861(119909) = ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
le ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861) and sdot sdot sdot and 119906
119862(119886119899)
le ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861otimes sdot sdot sdot otimes 119886
119899)
le 119906119862(119909)
V119861 (119909) = ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
ge ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861) or sdot sdot sdot or 119906
119862(119886119899)
ge ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861otimes sdot sdot sdot otimes 119886
119899)
ge 119906119862 (119909)
(5)
and hence 119861 sube 119862 Thus 119861 = ⟨119860⟩
Example 7 Let 119871 = 0 119886 119887 1 with 0 lt 119886 lt 119887 lt 1 Theoperations otimes and rarr are defined as
otimes 0 119886 119887 1
0 0 0 0 0
119886 0 0 119886 119886
119887 0 119886 119887 119887
1 0 119886 119887 1
rarr 0 119886 119887 1
0 1 1 1 1
119886 119886 1 1 1
119887 0 119886 1 1
1 0 119886 119887 1
(6)
Define 119906119860 V119860
119871 rarr [0 1] as 119906119860(0) = 02 V
119860(0) =
07 119906119860(119886) = 05 V
119860(119886) = 045 119906
119860(119887) = 06 V
119860(119887) =
04 119906119860(1) = 08 and V
119860(1) = 015 Since 05 = 119906
119860(119886) and
119906119860(119886) lt 119906
119860(1198862) = 02 119891 is not a fuzzy filter It is routine
to verify that ⟨119860⟩ is an intuitionistic fuzzy filter where119906⟨119860⟩(0) = 119906
⟨119860⟩(119886) = 05 V
⟨119860⟩(0) = V
⟨119860⟩(119886) = 04 119906
⟨119860⟩(119887) =
06 V⟨119860⟩(119887) = 03 119906
⟨119860⟩(1) = 08 and V
⟨119860⟩(1) = 015 from the
above theorem
Lemma 8 Let 119886 119887 119904 119905 isin [0 1] such that 0 le 119886 + 119887 le 1 and0 le 119904 + 119905 le 1 Then 0 le 119886 or 119904 + 119887 and 119905 le 1
Proof Not losing the generality we assume that 119886 le 119904 Then119886 or 119904 + 119887 and 119905 le 119904 + 119905 le 1 It is obvious that 0 le 119886 or 119904 + 119887 and 119905Thus it holds that 0 le 119886 or 119904 + 119887 and 119905 le 1
Theorem 9 Let 119860 be an intuitionistic fuzzy filter of 119871 and forall 119905 119904 isin [0 1] such that 0 le 119905 + 119904 le 1 Then 119860119905119904
= (119906119905
119860 V119904119860) is
an intuitionistic fuzzy filter where
119906119905
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119905 119909 = 1
V119904119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and 119904 119909 = 1
(7)
Proof It follows from Lemma 8 that 119860119905119904isin IFS(119871) Now we
prove that 119860119905119904 is an intuitionistic fuzzy filterIf 119909 le 119910 we consider the following two cases
Case 1 (119910 = 1) It is obvious that 119906119905119860(119909) le 119906
119905
119860(1) = 119906
119905
119860(119910)
V119905119860(119909) ge V119905
119860(1) = V119905
119860(119910)
Case 2 (119910 = 1) The definition of 119860119905119904 leads that 119906119905119860(119909) =
119906119860(119909) le 119906
119860(119910) = 119906
119905
119860(119910) V119905
119860(119909) = V
119860(119909) ge V
119860(119910) = V119905
119860(119910)
Thus 119906119905119860(119909) le 119906
119905
119860(119910) V119905
119860(119909) ge V119905
119860(119910)
Let 119909 119910 isin 119871 We consider the following two cases
Case 1 (119909 otimes 119910 = 1) If 119909 = 119910 = 1 it is obvious that 119906119905119860(119909) and
119906119905
119860(119910) le 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) ge V119905
119860(119909 otimes 119910)
If 119909 = 1 119910 = 1 or 119909 = 1 119910 = 1 it is a contradictionIf 119909 = 1 119910 = 1 it holds that 119906119905
119860(119909) and 119906
119905
119860(119910) = 119906
119860(119909) and
119906119860(119910) le 119906
119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or
119910119860(119909) ge V
119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
Case 2 (119909 otimes 119910 = 1) If 119909 = 119910 = 1 it is a contradictionIf 119909 = 1 119910 = 1 or 119909 = 1 119910 = 1 it is obvious that 119906119905
119860(119909) and
119906119905
119860(119910) le 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) ge V119905
119860(119909 otimes 119910)
If 119909 = 1 119910 = 1we have 119906119905119860(119909) and 119906
119905
119860(119910) = 119906
119860(119909) and 119906
119860(119910) le
119906119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or 119910
119860(119909) ge
V119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
All in all it yields that 119906119905119860(119909)and119906
119905
119860(119910) = 119906
119860(119909)and119906
119860(119910) le
119906119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or 119910
119860(119909) ge
V119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
Thus 119860119905119904 is an intuitionistic fuzzy filter
For given 119860 119861 isin IFS(119871) the operation ⋆ is defined by
119860⋆119861 = (119906119860otimes119906
119861 V119860⊙V
119861) (8)
where
119906119860otimes119906
119861(119909) = ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
V119860⊙V
119861(119909) = ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
(9)
Furthermore the tip-extended pair for intuitionisticfuzzy sets 119860 and 119861 are defined by
119860119861= (119906
119906119861
119860 VV119861119860) (10)
4 Journal of Applied Mathematics
where
119906119906119861
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119906119861 (1) 119909 = 1
VV119861119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and V
119861(1) 119909 = 1
119861119860= (119906
119906119860
119861 VV119860119861)
(11)
where
119906119906119860
119861(119909) =
119906119861(119909) 119909 = 1
119906119861 (1) or 119906119860 (1) 119909 = 1
VV119860119861(119909) =
V119861 (119909) 119909 = 1
V119861(1) and V
119860(1) 119909 = 1
(12)
Theorem 10 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860isin 119868119865119865(119871)
Proof It is obvious that 119906119906119861119860otimes119906
119906119860
119861is order-preserving and
VV119861119860⊙VV119860
119861is order-reserving For all 119909 119910 isin 119871 we have
119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910)
= ⋁
119906otimesVle119909otimes119910119906
119906119861
119860(119906) and 119906
119906119860
119861(V)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901 otimes 119903) and 119906
119906119860
119861(119902 otimes 119904)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901) and 119906
119906119861
119860(119903) and 119906
119906119860
119861(119902) and 119906
119906119860
119861(119904)
= ⋁
119901otimes119902le119909
119906119906119861
119860(119901) and 119906
119906119860
119861(119902) and ⋁
119903otimes119904le119910
119906119906119861
119860(119903) and 119906
119906119860
119861(119904)
= 119906119906119861
119860otimes119906
119906119860
119861(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119910)
VV119861119860⊙VV119860
119861(119909 otimes 119910)
= ⋀
119906otimesVle119909otimes119910VV119861119860(119906) or VV119860
119861(V)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901 otimes 119903) or VV119860
119861(119902 otimes 119904)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901) or VV119861
119860(119903) or VV119860
119861(119902) or VV119860
119861(119904)
= ⋀
119901otimes119902le119909
VV119861119860(119901) or VV119860
119861(119902) or ⋀
119903otimes119904le119910
VV119861119860(119903) or VV119860
119861(119904)
= VV119861119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
(13)
and hence 119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910) ge 119906
119906119861
119860otimes119906
119906119860
119861(119909) and
119906119906119861
119860otimes119906
119906119860
119861(119910) VV119861
119860⊙VV119860
119861(119909 otimes 119910) le VV119861
119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
Thus 119860119861⋆119861
119860isin IFF(119871)
Theorem 11 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Proof It is easy to prove that 119860 119861 sube 119860119861⋆119861
119860 and hence 119860 cup
119861 sube 119860119861⋆119861
119860 Thus ⟨119860 cup 119861⟩ sube 119860119861⋆119861
119860Assume that 119862 isin IFF(119871) such that 119860 cup 119861 sube 119862 If 119909 = 1
we have 119906119906119861119860otimes119906
119906119860
119861(1) = 119906
119860(1) or 119906
119861(1) le 119906
119862(1) VV119861
119860⊙VV119860
119861(1) =
V119860(1) and V
119861(1) ge V
119862(1) If 119909 lt 1 it holds that
119906119906119861
119860otimes119906
119906119860
119861(119909)
= ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119906119861
119860(119910) and 119906
119906119860
119861(119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119860(119910) and 119906
119861 (119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861 (119911)
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119910) and 119906
119862(119911) or ⋁
119910le119909
119906119862(119910) or ⋁
119911le119909
119906119862(119911)
= ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
le 119906119862(119909)
VV119861119860⊙VV119860
119861(119909)
= ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
VV119861119860(119910) or VV119860
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119860(119910) or V
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119910) or V
119862 (119911) and ⋀
119910le119909
V119862(119910) and ⋀
119911le119909
V119862 (119911)
= ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
ge V119862 (119909)
(14)
It follows fromTheorem 10 that 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Associating with the above results we prove the maintheorem here For 119860 119861 isin IFF(119871) the operations ⊓ and ⊔ onIFF(119871) are defined by
119860 ⊓ 119861 = 119860 cap 119861 119860 ⊔ 119861 = 119860119861⋆119861
119860 (15)
Theorem 12 (IFF(119871) ⊓ ⊔ 0 119871) is a bounded distributivelattice
Proof We only verify the distributivity Obviously it holdsthat 119862 ⊓ (119860 ⊔ 119861) supe (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) so we only prove
Journal of Applied Mathematics 5
that 119862 ⊓ (119860 ⊔ 119861) sube (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) Assume that 119909 isin 119871
for 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861(119909) le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860(119909) we
consider the following two cases
Case 1 (119909 = 1) We have
119906119862and 119906
119860⊔119861(1) = 119906
119862(1) and 119906
119906119861
119860otimes119906
119906119860
119861(1)
= 119906119862 (1) and (119906119860 (1) or 119906119861 (1))
= (119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(1)
V119862or V
119860⊔119861(1) = V
119862(1) or VV119861
119860⊙VV119860
119861(1)
= V119862 (1) or (V119860 (1) and V
119861 (1))
= (V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(1)
(16)
Case 2 (119909 = 1) It holds that
119906119862(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119909)
= 119906119862 (119909) and ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119906119862(119909) and 119906
119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862 (119909) and 119906
119906119861
119860(1) and 119906119861 (119909)
or 119906119862(119909) and 119906
119860(119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862 (119909) and 119906119860 (119910) and 119906119862 (119909) and 119906119861 (119911)
or 119906119862(119909) and 119906
119862(1) and 119906
119906119861
119860(1) and 119906
119861(119909)
or 119906119862 (119909) and 119906119862 (1) and 119906119860 (119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862(1) and (119906
119860(1) or 119906
119861(1))
and [(119906119862 (119909) and 119906119861 (119909)) or (119906119862 (119909) and 119906119860 (119909))]
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or [(119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))]
and [(119906119862(119909) and 119906
119861(119909)) or (119906
119862(119909) and 119906
119860(119909))]
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
or [(119906119862and 119906
119860)119906119862and119906119861
(1) and (119906119862and 119906
119860) (119910 or 119909)]
or [(119906119862and 119906
119861)119906119862and119906119860
(1) and (119906119862and 119906
119861) (119911 or 119909)]
= ⋁
119910otimes119911le119909
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
(17)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋁
1199101015840otimes1199111015840le119909
(119906119862and 119906
119860)119906119862and119906119861
(1199101015840) and (119906
119862and 119906
119861)119906119862and119906119860
(1199111015840)
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(119909)
V119862(119909) or VV119861
119860⊙VV119860
119861(119909)
= V119862 (119909) or ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
V119862(119909) or VV119861
119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) and V
119860(119909) and VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or V
119862(1) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) or V
119862(1) or V
119860(119909) or VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862 (119909) or V
119860(119910) or V
119862 (119909) or V119861 (119911)
and V119862 (1) or (V119860 (1) and V
119861 (1))
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and [(V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))]
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
6 Journal of Applied Mathematics
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
and [(V119862or V
119860)V119862orV119861
(1) or (V119862or V
119860) (119910 or 119909)]
and [(V119862or V
119861)V119862orV119860
(1) and (V119862or V
119861) (119911 or 119909)]
= ⋀
119910otimes119911le119909
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
(18)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋀
1199101015840otimes1199111015840le119909
(V119862or V
119860)V119862orV119861
(1199101015840) or (V
119862or V
119861)V119862orV119860
(1199111015840)
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(119909)
(19)
Thus 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860 and
V119862(119909) or VV119861
119860⊙VV119860
119861(119909) ge (V
119862or V
119860)V119862orV119861 ⊙(V
119862or V
119861)V119862orV119860 that is
the distributivity holds
4 Conclusions
In this paper by the notion of tip-extended pair of intuitionis-tic fuzzy sets it is verified that the set of all intuitionistic fuzzyfilters forms a bounded distributive lattice
Future research will focus on the lattice of fuzzy notionsin other algebras by tip-extended pair
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by AMEP of Linyi Universityand the Natural Science Foundation of Shandong Province(Grant no ZR2013FL006)
References
[1] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[2] F Esteva and L Godo ldquoMonoidal 119905-norm based logic towards alogic for left-continuous t-normsrdquo Fuzzy Sets and Systems vol124 no 3 pp 271ndash288 2001
[3] P Hajek Metamathematics of Fuzzy Logic Kluwer AcademicDordrecht The Netherlands 1998
[4] C C Chang ldquoAlgebraic analysis of many valued logicsrdquo Trans-actions of the American Mathematical Society vol 88 pp 467ndash490 1958
[5] G J Wang An Introduction to Mathematical Logic and Resolu-tion Principle Science in China Press Beijing China 2003
[6] E Turunen Mathematics Behind Fuzzy Logic Physica Heidel-berg Germany 1999
[7] B van Gasse G Deschrijver C Cornelis and E E KerreldquoFilters of residuated lattices and triangle algebrasrdquo InformationSciences vol 180 no 16 pp 3006ndash3020 2010
[8] E Turunen ldquoBoolean deductive systems of BL-algebrasrdquoArchive forMathematical Logic vol 40 no 6 pp 467ndash473 2001
[9] M Kondo ldquoFilters on commutative residuated latticesrdquo inIntegrated Uncertainty Management and Applications vol 68pp 343ndash347 Springer Berlin Germany 2010
[10] L Liu and K Li ldquoBoolean filters and positive implicative filtersof residuated latticesrdquo Information Sciences vol 177 no 24 pp5725ndash5738 2007
[11] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquoInformation Sciences vol 180 no 19 pp 3614ndash3632 2010
[12] MHaveshki A B Saeid and E Eslami ldquoSome types of filters in119861119871 algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006
[13] M Kondo andW A Dudek ldquoFilter theory of BL algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008
[14] A B Saeid and S Motamed ldquoNormal filters in BL-algebrasrdquoWorld Applied Sciences Journal vol 7 pp 70ndash76 2009
[15] E Turunen ldquoBL-algebras of basic fuzzy logicrdquoMathware amp SoftComputing vol 6 no 1 pp 49ndash61 1999
[16] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 no 3 pp 512ndash517 1971
[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
[18] J N Mordeson and D S Malik Fuzzy Commutative AlgebraWorld Scientific London UK 1998
[19] T Head ldquoA metatheorem for deriving fuzzy theorems fromcrisp versionsrdquo Fuzzy Sets and Systems vol 73 no 3 pp 349ndash358 1995
[20] T Head ldquoErratum to ldquoA metatheorem for deriving fuzzytheorems from crisp versionsrsquordquo Fuzzy Sets and Systems vol 79no 2 pp 277ndash278 1996
[21] I Jahan ldquoThe lattice of 119871-ideals of a ring is modularrdquo Fuzzy Setsand Systems vol 199 pp 121ndash129 2012
[22] Y B Jun Y Xu and X H Zhang ldquoFuzzy filters of MTL-algebrasrdquo Information Sciences vol 175 no 1-2 pp 120ndash1382005
[23] L Liu and K Li ldquoFuzzy filters of 119861119871-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005
[24] J L Zhang and H J Zhou ldquoFuzzy filters on the residuatedlatticesrdquo New Mathematics and Natural Compution vol 2 no1 pp 11ndash28 2006
[25] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[26] Z Xue Y XiaoW Liu H Cheng and Y Li ldquoIntuitionistic fuzzyfilter theory of 119861119871-algebrasrdquo International Journal of MachineLearning and Cybernetics vol 4 no 6 pp 659ndash669 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Secondly let 119862 be an intuitionistic fuzzy filter suchthat 119862 supe 119860 By the definition of intuitionistic fuzzy filter itholds that
119906119861(119909) = ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) and sdot sdot sdot and 119906
119860(119886119899)
le ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861) and sdot sdot sdot and 119906
119862(119886119899)
le ⋁
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861otimes sdot sdot sdot otimes 119886
119899)
le 119906119862(119909)
V119861 (119909) = ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119860(1198861) or sdot sdot sdot or 119906
119860(119886119899)
ge ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861) or sdot sdot sdot or 119906
119862(119886119899)
ge ⋀
1198861otimessdotsdotsdototimes119886
119899le119909
119886119894isin119871119899isin119873
119906119862(1198861otimes sdot sdot sdot otimes 119886
119899)
ge 119906119862 (119909)
(5)
and hence 119861 sube 119862 Thus 119861 = ⟨119860⟩
Example 7 Let 119871 = 0 119886 119887 1 with 0 lt 119886 lt 119887 lt 1 Theoperations otimes and rarr are defined as
otimes 0 119886 119887 1
0 0 0 0 0
119886 0 0 119886 119886
119887 0 119886 119887 119887
1 0 119886 119887 1
rarr 0 119886 119887 1
0 1 1 1 1
119886 119886 1 1 1
119887 0 119886 1 1
1 0 119886 119887 1
(6)
Define 119906119860 V119860
119871 rarr [0 1] as 119906119860(0) = 02 V
119860(0) =
07 119906119860(119886) = 05 V
119860(119886) = 045 119906
119860(119887) = 06 V
119860(119887) =
04 119906119860(1) = 08 and V
119860(1) = 015 Since 05 = 119906
119860(119886) and
119906119860(119886) lt 119906
119860(1198862) = 02 119891 is not a fuzzy filter It is routine
to verify that ⟨119860⟩ is an intuitionistic fuzzy filter where119906⟨119860⟩(0) = 119906
⟨119860⟩(119886) = 05 V
⟨119860⟩(0) = V
⟨119860⟩(119886) = 04 119906
⟨119860⟩(119887) =
06 V⟨119860⟩(119887) = 03 119906
⟨119860⟩(1) = 08 and V
⟨119860⟩(1) = 015 from the
above theorem
Lemma 8 Let 119886 119887 119904 119905 isin [0 1] such that 0 le 119886 + 119887 le 1 and0 le 119904 + 119905 le 1 Then 0 le 119886 or 119904 + 119887 and 119905 le 1
Proof Not losing the generality we assume that 119886 le 119904 Then119886 or 119904 + 119887 and 119905 le 119904 + 119905 le 1 It is obvious that 0 le 119886 or 119904 + 119887 and 119905Thus it holds that 0 le 119886 or 119904 + 119887 and 119905 le 1
Theorem 9 Let 119860 be an intuitionistic fuzzy filter of 119871 and forall 119905 119904 isin [0 1] such that 0 le 119905 + 119904 le 1 Then 119860119905119904
= (119906119905
119860 V119904119860) is
an intuitionistic fuzzy filter where
119906119905
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119905 119909 = 1
V119904119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and 119904 119909 = 1
(7)
Proof It follows from Lemma 8 that 119860119905119904isin IFS(119871) Now we
prove that 119860119905119904 is an intuitionistic fuzzy filterIf 119909 le 119910 we consider the following two cases
Case 1 (119910 = 1) It is obvious that 119906119905119860(119909) le 119906
119905
119860(1) = 119906
119905
119860(119910)
V119905119860(119909) ge V119905
119860(1) = V119905
119860(119910)
Case 2 (119910 = 1) The definition of 119860119905119904 leads that 119906119905119860(119909) =
119906119860(119909) le 119906
119860(119910) = 119906
119905
119860(119910) V119905
119860(119909) = V
119860(119909) ge V
119860(119910) = V119905
119860(119910)
Thus 119906119905119860(119909) le 119906
119905
119860(119910) V119905
119860(119909) ge V119905
119860(119910)
Let 119909 119910 isin 119871 We consider the following two cases
Case 1 (119909 otimes 119910 = 1) If 119909 = 119910 = 1 it is obvious that 119906119905119860(119909) and
119906119905
119860(119910) le 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) ge V119905
119860(119909 otimes 119910)
If 119909 = 1 119910 = 1 or 119909 = 1 119910 = 1 it is a contradictionIf 119909 = 1 119910 = 1 it holds that 119906119905
119860(119909) and 119906
119905
119860(119910) = 119906
119860(119909) and
119906119860(119910) le 119906
119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or
119910119860(119909) ge V
119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
Case 2 (119909 otimes 119910 = 1) If 119909 = 119910 = 1 it is a contradictionIf 119909 = 1 119910 = 1 or 119909 = 1 119910 = 1 it is obvious that 119906119905
119860(119909) and
119906119905
119860(119910) le 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) ge V119905
119860(119909 otimes 119910)
If 119909 = 1 119910 = 1we have 119906119905119860(119909) and 119906
119905
119860(119910) = 119906
119860(119909) and 119906
119860(119910) le
119906119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or 119910
119860(119909) ge
V119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
All in all it yields that 119906119905119860(119909)and119906
119905
119860(119910) = 119906
119860(119909)and119906
119860(119910) le
119906119860(119909 otimes 119910) = 119906
119905
119860(119909 otimes 119910) V119905
119860(119909) or 119910
119905
119860(119909) = V
119860(119909) or 119910
119860(119909) ge
V119860(119909 otimes 119910) = V119905
119860(119909 otimes 119910)
Thus 119860119905119904 is an intuitionistic fuzzy filter
For given 119860 119861 isin IFS(119871) the operation ⋆ is defined by
119860⋆119861 = (119906119860otimes119906
119861 V119860⊙V
119861) (8)
where
119906119860otimes119906
119861(119909) = ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
V119860⊙V
119861(119909) = ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
(9)
Furthermore the tip-extended pair for intuitionisticfuzzy sets 119860 and 119861 are defined by
119860119861= (119906
119906119861
119860 VV119861119860) (10)
4 Journal of Applied Mathematics
where
119906119906119861
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119906119861 (1) 119909 = 1
VV119861119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and V
119861(1) 119909 = 1
119861119860= (119906
119906119860
119861 VV119860119861)
(11)
where
119906119906119860
119861(119909) =
119906119861(119909) 119909 = 1
119906119861 (1) or 119906119860 (1) 119909 = 1
VV119860119861(119909) =
V119861 (119909) 119909 = 1
V119861(1) and V
119860(1) 119909 = 1
(12)
Theorem 10 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860isin 119868119865119865(119871)
Proof It is obvious that 119906119906119861119860otimes119906
119906119860
119861is order-preserving and
VV119861119860⊙VV119860
119861is order-reserving For all 119909 119910 isin 119871 we have
119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910)
= ⋁
119906otimesVle119909otimes119910119906
119906119861
119860(119906) and 119906
119906119860
119861(V)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901 otimes 119903) and 119906
119906119860
119861(119902 otimes 119904)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901) and 119906
119906119861
119860(119903) and 119906
119906119860
119861(119902) and 119906
119906119860
119861(119904)
= ⋁
119901otimes119902le119909
119906119906119861
119860(119901) and 119906
119906119860
119861(119902) and ⋁
119903otimes119904le119910
119906119906119861
119860(119903) and 119906
119906119860
119861(119904)
= 119906119906119861
119860otimes119906
119906119860
119861(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119910)
VV119861119860⊙VV119860
119861(119909 otimes 119910)
= ⋀
119906otimesVle119909otimes119910VV119861119860(119906) or VV119860
119861(V)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901 otimes 119903) or VV119860
119861(119902 otimes 119904)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901) or VV119861
119860(119903) or VV119860
119861(119902) or VV119860
119861(119904)
= ⋀
119901otimes119902le119909
VV119861119860(119901) or VV119860
119861(119902) or ⋀
119903otimes119904le119910
VV119861119860(119903) or VV119860
119861(119904)
= VV119861119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
(13)
and hence 119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910) ge 119906
119906119861
119860otimes119906
119906119860
119861(119909) and
119906119906119861
119860otimes119906
119906119860
119861(119910) VV119861
119860⊙VV119860
119861(119909 otimes 119910) le VV119861
119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
Thus 119860119861⋆119861
119860isin IFF(119871)
Theorem 11 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Proof It is easy to prove that 119860 119861 sube 119860119861⋆119861
119860 and hence 119860 cup
119861 sube 119860119861⋆119861
119860 Thus ⟨119860 cup 119861⟩ sube 119860119861⋆119861
119860Assume that 119862 isin IFF(119871) such that 119860 cup 119861 sube 119862 If 119909 = 1
we have 119906119906119861119860otimes119906
119906119860
119861(1) = 119906
119860(1) or 119906
119861(1) le 119906
119862(1) VV119861
119860⊙VV119860
119861(1) =
V119860(1) and V
119861(1) ge V
119862(1) If 119909 lt 1 it holds that
119906119906119861
119860otimes119906
119906119860
119861(119909)
= ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119906119861
119860(119910) and 119906
119906119860
119861(119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119860(119910) and 119906
119861 (119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861 (119911)
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119910) and 119906
119862(119911) or ⋁
119910le119909
119906119862(119910) or ⋁
119911le119909
119906119862(119911)
= ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
le 119906119862(119909)
VV119861119860⊙VV119860
119861(119909)
= ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
VV119861119860(119910) or VV119860
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119860(119910) or V
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119910) or V
119862 (119911) and ⋀
119910le119909
V119862(119910) and ⋀
119911le119909
V119862 (119911)
= ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
ge V119862 (119909)
(14)
It follows fromTheorem 10 that 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Associating with the above results we prove the maintheorem here For 119860 119861 isin IFF(119871) the operations ⊓ and ⊔ onIFF(119871) are defined by
119860 ⊓ 119861 = 119860 cap 119861 119860 ⊔ 119861 = 119860119861⋆119861
119860 (15)
Theorem 12 (IFF(119871) ⊓ ⊔ 0 119871) is a bounded distributivelattice
Proof We only verify the distributivity Obviously it holdsthat 119862 ⊓ (119860 ⊔ 119861) supe (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) so we only prove
Journal of Applied Mathematics 5
that 119862 ⊓ (119860 ⊔ 119861) sube (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) Assume that 119909 isin 119871
for 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861(119909) le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860(119909) we
consider the following two cases
Case 1 (119909 = 1) We have
119906119862and 119906
119860⊔119861(1) = 119906
119862(1) and 119906
119906119861
119860otimes119906
119906119860
119861(1)
= 119906119862 (1) and (119906119860 (1) or 119906119861 (1))
= (119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(1)
V119862or V
119860⊔119861(1) = V
119862(1) or VV119861
119860⊙VV119860
119861(1)
= V119862 (1) or (V119860 (1) and V
119861 (1))
= (V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(1)
(16)
Case 2 (119909 = 1) It holds that
119906119862(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119909)
= 119906119862 (119909) and ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119906119862(119909) and 119906
119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862 (119909) and 119906
119906119861
119860(1) and 119906119861 (119909)
or 119906119862(119909) and 119906
119860(119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862 (119909) and 119906119860 (119910) and 119906119862 (119909) and 119906119861 (119911)
or 119906119862(119909) and 119906
119862(1) and 119906
119906119861
119860(1) and 119906
119861(119909)
or 119906119862 (119909) and 119906119862 (1) and 119906119860 (119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862(1) and (119906
119860(1) or 119906
119861(1))
and [(119906119862 (119909) and 119906119861 (119909)) or (119906119862 (119909) and 119906119860 (119909))]
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or [(119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))]
and [(119906119862(119909) and 119906
119861(119909)) or (119906
119862(119909) and 119906
119860(119909))]
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
or [(119906119862and 119906
119860)119906119862and119906119861
(1) and (119906119862and 119906
119860) (119910 or 119909)]
or [(119906119862and 119906
119861)119906119862and119906119860
(1) and (119906119862and 119906
119861) (119911 or 119909)]
= ⋁
119910otimes119911le119909
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
(17)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋁
1199101015840otimes1199111015840le119909
(119906119862and 119906
119860)119906119862and119906119861
(1199101015840) and (119906
119862and 119906
119861)119906119862and119906119860
(1199111015840)
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(119909)
V119862(119909) or VV119861
119860⊙VV119860
119861(119909)
= V119862 (119909) or ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
V119862(119909) or VV119861
119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) and V
119860(119909) and VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or V
119862(1) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) or V
119862(1) or V
119860(119909) or VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862 (119909) or V
119860(119910) or V
119862 (119909) or V119861 (119911)
and V119862 (1) or (V119860 (1) and V
119861 (1))
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and [(V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))]
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
6 Journal of Applied Mathematics
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
and [(V119862or V
119860)V119862orV119861
(1) or (V119862or V
119860) (119910 or 119909)]
and [(V119862or V
119861)V119862orV119860
(1) and (V119862or V
119861) (119911 or 119909)]
= ⋀
119910otimes119911le119909
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
(18)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋀
1199101015840otimes1199111015840le119909
(V119862or V
119860)V119862orV119861
(1199101015840) or (V
119862or V
119861)V119862orV119860
(1199111015840)
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(119909)
(19)
Thus 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860 and
V119862(119909) or VV119861
119860⊙VV119860
119861(119909) ge (V
119862or V
119860)V119862orV119861 ⊙(V
119862or V
119861)V119862orV119860 that is
the distributivity holds
4 Conclusions
In this paper by the notion of tip-extended pair of intuitionis-tic fuzzy sets it is verified that the set of all intuitionistic fuzzyfilters forms a bounded distributive lattice
Future research will focus on the lattice of fuzzy notionsin other algebras by tip-extended pair
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by AMEP of Linyi Universityand the Natural Science Foundation of Shandong Province(Grant no ZR2013FL006)
References
[1] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[2] F Esteva and L Godo ldquoMonoidal 119905-norm based logic towards alogic for left-continuous t-normsrdquo Fuzzy Sets and Systems vol124 no 3 pp 271ndash288 2001
[3] P Hajek Metamathematics of Fuzzy Logic Kluwer AcademicDordrecht The Netherlands 1998
[4] C C Chang ldquoAlgebraic analysis of many valued logicsrdquo Trans-actions of the American Mathematical Society vol 88 pp 467ndash490 1958
[5] G J Wang An Introduction to Mathematical Logic and Resolu-tion Principle Science in China Press Beijing China 2003
[6] E Turunen Mathematics Behind Fuzzy Logic Physica Heidel-berg Germany 1999
[7] B van Gasse G Deschrijver C Cornelis and E E KerreldquoFilters of residuated lattices and triangle algebrasrdquo InformationSciences vol 180 no 16 pp 3006ndash3020 2010
[8] E Turunen ldquoBoolean deductive systems of BL-algebrasrdquoArchive forMathematical Logic vol 40 no 6 pp 467ndash473 2001
[9] M Kondo ldquoFilters on commutative residuated latticesrdquo inIntegrated Uncertainty Management and Applications vol 68pp 343ndash347 Springer Berlin Germany 2010
[10] L Liu and K Li ldquoBoolean filters and positive implicative filtersof residuated latticesrdquo Information Sciences vol 177 no 24 pp5725ndash5738 2007
[11] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquoInformation Sciences vol 180 no 19 pp 3614ndash3632 2010
[12] MHaveshki A B Saeid and E Eslami ldquoSome types of filters in119861119871 algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006
[13] M Kondo andW A Dudek ldquoFilter theory of BL algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008
[14] A B Saeid and S Motamed ldquoNormal filters in BL-algebrasrdquoWorld Applied Sciences Journal vol 7 pp 70ndash76 2009
[15] E Turunen ldquoBL-algebras of basic fuzzy logicrdquoMathware amp SoftComputing vol 6 no 1 pp 49ndash61 1999
[16] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 no 3 pp 512ndash517 1971
[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
[18] J N Mordeson and D S Malik Fuzzy Commutative AlgebraWorld Scientific London UK 1998
[19] T Head ldquoA metatheorem for deriving fuzzy theorems fromcrisp versionsrdquo Fuzzy Sets and Systems vol 73 no 3 pp 349ndash358 1995
[20] T Head ldquoErratum to ldquoA metatheorem for deriving fuzzytheorems from crisp versionsrsquordquo Fuzzy Sets and Systems vol 79no 2 pp 277ndash278 1996
[21] I Jahan ldquoThe lattice of 119871-ideals of a ring is modularrdquo Fuzzy Setsand Systems vol 199 pp 121ndash129 2012
[22] Y B Jun Y Xu and X H Zhang ldquoFuzzy filters of MTL-algebrasrdquo Information Sciences vol 175 no 1-2 pp 120ndash1382005
[23] L Liu and K Li ldquoFuzzy filters of 119861119871-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005
[24] J L Zhang and H J Zhou ldquoFuzzy filters on the residuatedlatticesrdquo New Mathematics and Natural Compution vol 2 no1 pp 11ndash28 2006
[25] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[26] Z Xue Y XiaoW Liu H Cheng and Y Li ldquoIntuitionistic fuzzyfilter theory of 119861119871-algebrasrdquo International Journal of MachineLearning and Cybernetics vol 4 no 6 pp 659ndash669 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
where
119906119906119861
119860(119909) =
119906119860(119909) 119909 = 1
119906119860 (1) or 119906119861 (1) 119909 = 1
VV119861119860(119909) =
V119860(119909) 119909 = 1
V119860(1) and V
119861(1) 119909 = 1
119861119860= (119906
119906119860
119861 VV119860119861)
(11)
where
119906119906119860
119861(119909) =
119906119861(119909) 119909 = 1
119906119861 (1) or 119906119860 (1) 119909 = 1
VV119860119861(119909) =
V119861 (119909) 119909 = 1
V119861(1) and V
119860(1) 119909 = 1
(12)
Theorem 10 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860isin 119868119865119865(119871)
Proof It is obvious that 119906119906119861119860otimes119906
119906119860
119861is order-preserving and
VV119861119860⊙VV119860
119861is order-reserving For all 119909 119910 isin 119871 we have
119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910)
= ⋁
119906otimesVle119909otimes119910119906
119906119861
119860(119906) and 119906
119906119860
119861(V)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901 otimes 119903) and 119906
119906119860
119861(119902 otimes 119904)
ge ⋁
119901otimes119902le119909
119903otimes119904le119910
119906119906119861
119860(119901) and 119906
119906119861
119860(119903) and 119906
119906119860
119861(119902) and 119906
119906119860
119861(119904)
= ⋁
119901otimes119902le119909
119906119906119861
119860(119901) and 119906
119906119860
119861(119902) and ⋁
119903otimes119904le119910
119906119906119861
119860(119903) and 119906
119906119860
119861(119904)
= 119906119906119861
119860otimes119906
119906119860
119861(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119910)
VV119861119860⊙VV119860
119861(119909 otimes 119910)
= ⋀
119906otimesVle119909otimes119910VV119861119860(119906) or VV119860
119861(V)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901 otimes 119903) or VV119860
119861(119902 otimes 119904)
le ⋀
119901otimes119902le119909
119903otimes119904le119910
VV119861119860(119901) or VV119861
119860(119903) or VV119860
119861(119902) or VV119860
119861(119904)
= ⋀
119901otimes119902le119909
VV119861119860(119901) or VV119860
119861(119902) or ⋀
119903otimes119904le119910
VV119861119860(119903) or VV119860
119861(119904)
= VV119861119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
(13)
and hence 119906119906119861
119860otimes119906
119906119860
119861(119909 otimes 119910) ge 119906
119906119861
119860otimes119906
119906119860
119861(119909) and
119906119906119861
119860otimes119906
119906119860
119861(119910) VV119861
119860⊙VV119860
119861(119909 otimes 119910) le VV119861
119860⊙VV119860
119861(119909) or VV119861
119860⊙VV119860
119861(119910)
Thus 119860119861⋆119861
119860isin IFF(119871)
Theorem 11 Let 119860 119861 isin 119868119865119865(119871) Then 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Proof It is easy to prove that 119860 119861 sube 119860119861⋆119861
119860 and hence 119860 cup
119861 sube 119860119861⋆119861
119860 Thus ⟨119860 cup 119861⟩ sube 119860119861⋆119861
119860Assume that 119862 isin IFF(119871) such that 119860 cup 119861 sube 119862 If 119909 = 1
we have 119906119906119861119860otimes119906
119906119860
119861(1) = 119906
119860(1) or 119906
119861(1) le 119906
119862(1) VV119861
119860⊙VV119860
119861(1) =
V119860(1) and V
119861(1) ge V
119862(1) If 119909 lt 1 it holds that
119906119906119861
119860otimes119906
119906119860
119861(119909)
= ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119906119861
119860(119910) and 119906
119906119860
119861(119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119860(119910) and 119906
119861 (119911) or ⋁
119910le119909
119906119860(119910) or ⋁
119911le119909
119906119861 (119911)
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119910) and 119906
119862(119911) or ⋁
119910le119909
119906119862(119910) or ⋁
119911le119909
119906119862(119911)
= ⋁
119910otimes119911le119909
119906119860(119910) and 119906
119861(119911)
le 119906119862(119909)
VV119861119860⊙VV119860
119861(119909)
= ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
VV119861119860(119910) or VV119860
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119860(119910) or V
119861(119911) and ⋀
119910le119909
V119860(119910) and ⋀
119911le119909
V119861(119911)
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119910) or V
119862 (119911) and ⋀
119910le119909
V119862(119910) and ⋀
119911le119909
V119862 (119911)
= ⋀
119910otimes119911le119909
V119860(119910) or V
119861(119911)
ge V119862 (119909)
(14)
It follows fromTheorem 10 that 119860119861⋆119861
119860= ⟨119860 cup 119861⟩
Associating with the above results we prove the maintheorem here For 119860 119861 isin IFF(119871) the operations ⊓ and ⊔ onIFF(119871) are defined by
119860 ⊓ 119861 = 119860 cap 119861 119860 ⊔ 119861 = 119860119861⋆119861
119860 (15)
Theorem 12 (IFF(119871) ⊓ ⊔ 0 119871) is a bounded distributivelattice
Proof We only verify the distributivity Obviously it holdsthat 119862 ⊓ (119860 ⊔ 119861) supe (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) so we only prove
Journal of Applied Mathematics 5
that 119862 ⊓ (119860 ⊔ 119861) sube (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) Assume that 119909 isin 119871
for 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861(119909) le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860(119909) we
consider the following two cases
Case 1 (119909 = 1) We have
119906119862and 119906
119860⊔119861(1) = 119906
119862(1) and 119906
119906119861
119860otimes119906
119906119860
119861(1)
= 119906119862 (1) and (119906119860 (1) or 119906119861 (1))
= (119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(1)
V119862or V
119860⊔119861(1) = V
119862(1) or VV119861
119860⊙VV119860
119861(1)
= V119862 (1) or (V119860 (1) and V
119861 (1))
= (V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(1)
(16)
Case 2 (119909 = 1) It holds that
119906119862(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119909)
= 119906119862 (119909) and ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119906119862(119909) and 119906
119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862 (119909) and 119906
119906119861
119860(1) and 119906119861 (119909)
or 119906119862(119909) and 119906
119860(119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862 (119909) and 119906119860 (119910) and 119906119862 (119909) and 119906119861 (119911)
or 119906119862(119909) and 119906
119862(1) and 119906
119906119861
119860(1) and 119906
119861(119909)
or 119906119862 (119909) and 119906119862 (1) and 119906119860 (119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862(1) and (119906
119860(1) or 119906
119861(1))
and [(119906119862 (119909) and 119906119861 (119909)) or (119906119862 (119909) and 119906119860 (119909))]
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or [(119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))]
and [(119906119862(119909) and 119906
119861(119909)) or (119906
119862(119909) and 119906
119860(119909))]
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
or [(119906119862and 119906
119860)119906119862and119906119861
(1) and (119906119862and 119906
119860) (119910 or 119909)]
or [(119906119862and 119906
119861)119906119862and119906119860
(1) and (119906119862and 119906
119861) (119911 or 119909)]
= ⋁
119910otimes119911le119909
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
(17)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋁
1199101015840otimes1199111015840le119909
(119906119862and 119906
119860)119906119862and119906119861
(1199101015840) and (119906
119862and 119906
119861)119906119862and119906119860
(1199111015840)
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(119909)
V119862(119909) or VV119861
119860⊙VV119860
119861(119909)
= V119862 (119909) or ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
V119862(119909) or VV119861
119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) and V
119860(119909) and VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or V
119862(1) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) or V
119862(1) or V
119860(119909) or VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862 (119909) or V
119860(119910) or V
119862 (119909) or V119861 (119911)
and V119862 (1) or (V119860 (1) and V
119861 (1))
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and [(V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))]
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
6 Journal of Applied Mathematics
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
and [(V119862or V
119860)V119862orV119861
(1) or (V119862or V
119860) (119910 or 119909)]
and [(V119862or V
119861)V119862orV119860
(1) and (V119862or V
119861) (119911 or 119909)]
= ⋀
119910otimes119911le119909
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
(18)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋀
1199101015840otimes1199111015840le119909
(V119862or V
119860)V119862orV119861
(1199101015840) or (V
119862or V
119861)V119862orV119860
(1199111015840)
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(119909)
(19)
Thus 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860 and
V119862(119909) or VV119861
119860⊙VV119860
119861(119909) ge (V
119862or V
119860)V119862orV119861 ⊙(V
119862or V
119861)V119862orV119860 that is
the distributivity holds
4 Conclusions
In this paper by the notion of tip-extended pair of intuitionis-tic fuzzy sets it is verified that the set of all intuitionistic fuzzyfilters forms a bounded distributive lattice
Future research will focus on the lattice of fuzzy notionsin other algebras by tip-extended pair
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by AMEP of Linyi Universityand the Natural Science Foundation of Shandong Province(Grant no ZR2013FL006)
References
[1] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[2] F Esteva and L Godo ldquoMonoidal 119905-norm based logic towards alogic for left-continuous t-normsrdquo Fuzzy Sets and Systems vol124 no 3 pp 271ndash288 2001
[3] P Hajek Metamathematics of Fuzzy Logic Kluwer AcademicDordrecht The Netherlands 1998
[4] C C Chang ldquoAlgebraic analysis of many valued logicsrdquo Trans-actions of the American Mathematical Society vol 88 pp 467ndash490 1958
[5] G J Wang An Introduction to Mathematical Logic and Resolu-tion Principle Science in China Press Beijing China 2003
[6] E Turunen Mathematics Behind Fuzzy Logic Physica Heidel-berg Germany 1999
[7] B van Gasse G Deschrijver C Cornelis and E E KerreldquoFilters of residuated lattices and triangle algebrasrdquo InformationSciences vol 180 no 16 pp 3006ndash3020 2010
[8] E Turunen ldquoBoolean deductive systems of BL-algebrasrdquoArchive forMathematical Logic vol 40 no 6 pp 467ndash473 2001
[9] M Kondo ldquoFilters on commutative residuated latticesrdquo inIntegrated Uncertainty Management and Applications vol 68pp 343ndash347 Springer Berlin Germany 2010
[10] L Liu and K Li ldquoBoolean filters and positive implicative filtersof residuated latticesrdquo Information Sciences vol 177 no 24 pp5725ndash5738 2007
[11] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquoInformation Sciences vol 180 no 19 pp 3614ndash3632 2010
[12] MHaveshki A B Saeid and E Eslami ldquoSome types of filters in119861119871 algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006
[13] M Kondo andW A Dudek ldquoFilter theory of BL algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008
[14] A B Saeid and S Motamed ldquoNormal filters in BL-algebrasrdquoWorld Applied Sciences Journal vol 7 pp 70ndash76 2009
[15] E Turunen ldquoBL-algebras of basic fuzzy logicrdquoMathware amp SoftComputing vol 6 no 1 pp 49ndash61 1999
[16] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 no 3 pp 512ndash517 1971
[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
[18] J N Mordeson and D S Malik Fuzzy Commutative AlgebraWorld Scientific London UK 1998
[19] T Head ldquoA metatheorem for deriving fuzzy theorems fromcrisp versionsrdquo Fuzzy Sets and Systems vol 73 no 3 pp 349ndash358 1995
[20] T Head ldquoErratum to ldquoA metatheorem for deriving fuzzytheorems from crisp versionsrsquordquo Fuzzy Sets and Systems vol 79no 2 pp 277ndash278 1996
[21] I Jahan ldquoThe lattice of 119871-ideals of a ring is modularrdquo Fuzzy Setsand Systems vol 199 pp 121ndash129 2012
[22] Y B Jun Y Xu and X H Zhang ldquoFuzzy filters of MTL-algebrasrdquo Information Sciences vol 175 no 1-2 pp 120ndash1382005
[23] L Liu and K Li ldquoFuzzy filters of 119861119871-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005
[24] J L Zhang and H J Zhou ldquoFuzzy filters on the residuatedlatticesrdquo New Mathematics and Natural Compution vol 2 no1 pp 11ndash28 2006
[25] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[26] Z Xue Y XiaoW Liu H Cheng and Y Li ldquoIntuitionistic fuzzyfilter theory of 119861119871-algebrasrdquo International Journal of MachineLearning and Cybernetics vol 4 no 6 pp 659ndash669 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
that 119862 ⊓ (119860 ⊔ 119861) sube (119862 ⊓ 119860) ⊔ (119862 ⊓ 119861) Assume that 119909 isin 119871
for 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861(119909) le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860(119909) we
consider the following two cases
Case 1 (119909 = 1) We have
119906119862and 119906
119860⊔119861(1) = 119906
119862(1) and 119906
119906119861
119860otimes119906
119906119860
119861(1)
= 119906119862 (1) and (119906119860 (1) or 119906119861 (1))
= (119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(1)
V119862or V
119860⊔119861(1) = V
119862(1) or VV119861
119860⊙VV119860
119861(1)
= V119862 (1) or (V119860 (1) and V
119861 (1))
= (V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(1)
(16)
Case 2 (119909 = 1) It holds that
119906119862(119909) and 119906
119906119861
119860otimes119906
119906119860
119861(119909)
= 119906119862 (119909) and ⋁
119910otimes119911le119909
119906119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119906119862(119909) and 119906
119906119861
119860(119910) and 119906
119906119860
119861(119911)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862 (119909) and 119906
119906119861
119860(1) and 119906119861 (119909)
or 119906119862(119909) and 119906
119860(119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862 (119909) and 119906119860 (119910) and 119906119862 (119909) and 119906119861 (119911)
or 119906119862(119909) and 119906
119862(1) and 119906
119906119861
119860(1) and 119906
119861(119909)
or 119906119862 (119909) and 119906119862 (1) and 119906119860 (119909) and 119906
119906119860
119861(1)
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or 119906119862(1) and (119906
119860(1) or 119906
119861(1))
and [(119906119862 (119909) and 119906119861 (119909)) or (119906119862 (119909) and 119906119860 (119909))]
= ⋁
119910otimes119911le119909
119910 = 1119911 = 1
119906119862(119909) and 119906
119860(119910) and 119906
119862(119909) and 119906
119861(119911)
or [(119906119862(1) and 119906
119860(1)) or (119906
119862(1) and 119906
119861(1))]
and [(119906119862(119909) and 119906
119861(119909)) or (119906
119862(119909) and 119906
119860(119909))]
le ⋁
119910otimes119911le119909
119910 = 1119911 = 1
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
or [(119906119862and 119906
119860)119906119862and119906119861
(1) and (119906119862and 119906
119860) (119910 or 119909)]
or [(119906119862and 119906
119861)119906119862and119906119860
(1) and (119906119862and 119906
119861) (119911 or 119909)]
= ⋁
119910otimes119911le119909
(119906119862and 119906
119860)119906119862and119906119861
(119910 or 119909)
and(119906119862and 119906
119861)119906119862and119906119860
(119911 or 119909)
(17)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋁
1199101015840otimes1199111015840le119909
(119906119862and 119906
119860)119906119862and119906119861
(1199101015840) and (119906
119862and 119906
119861)119906119862and119906119860
(1199111015840)
= (119906119862and 119906
119860)119906119862and119906119861
otimes(119906119862and 119906
119861)119906119862and119906119860
(119909)
V119862(119909) or VV119861
119860⊙VV119860
119861(119909)
= V119862 (119909) or ⋀
119910otimes119911le119909
VV119861119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
V119862(119909) or VV119861
119860(119910) or VV119860
119861(119911)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) and V
119860(119909) and VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and V119862(119909) or V
119862(1) or VV119861
119860(1) or V
119861(119909)
and V119862(119909) or V
119862(1) or V
119860(119909) or VV119860
119861(1)
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862 (119909) or V
119860(119910) or V
119862 (119909) or V119861 (119911)
and V119862 (1) or (V119860 (1) and V
119861 (1))
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
= ⋀
119910otimes119911le119909
119910 = 1119911 = 1
V119862(119909) or V
119860(119910) or V
119862(119909) or V
119861(119911)
and [(V119862(1) or V
119860(1)) and (V
119862(1) or V
119861(1))]
or [(V119862(119909) or V
119861(119909)) and (V
119862(119909) or V
119860(119909))]
6 Journal of Applied Mathematics
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
and [(V119862or V
119860)V119862orV119861
(1) or (V119862or V
119860) (119910 or 119909)]
and [(V119862or V
119861)V119862orV119860
(1) and (V119862or V
119861) (119911 or 119909)]
= ⋀
119910otimes119911le119909
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
(18)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋀
1199101015840otimes1199111015840le119909
(V119862or V
119860)V119862orV119861
(1199101015840) or (V
119862or V
119861)V119862orV119860
(1199111015840)
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(119909)
(19)
Thus 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860 and
V119862(119909) or VV119861
119860⊙VV119860
119861(119909) ge (V
119862or V
119860)V119862orV119861 ⊙(V
119862or V
119861)V119862orV119860 that is
the distributivity holds
4 Conclusions
In this paper by the notion of tip-extended pair of intuitionis-tic fuzzy sets it is verified that the set of all intuitionistic fuzzyfilters forms a bounded distributive lattice
Future research will focus on the lattice of fuzzy notionsin other algebras by tip-extended pair
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by AMEP of Linyi Universityand the Natural Science Foundation of Shandong Province(Grant no ZR2013FL006)
References
[1] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[2] F Esteva and L Godo ldquoMonoidal 119905-norm based logic towards alogic for left-continuous t-normsrdquo Fuzzy Sets and Systems vol124 no 3 pp 271ndash288 2001
[3] P Hajek Metamathematics of Fuzzy Logic Kluwer AcademicDordrecht The Netherlands 1998
[4] C C Chang ldquoAlgebraic analysis of many valued logicsrdquo Trans-actions of the American Mathematical Society vol 88 pp 467ndash490 1958
[5] G J Wang An Introduction to Mathematical Logic and Resolu-tion Principle Science in China Press Beijing China 2003
[6] E Turunen Mathematics Behind Fuzzy Logic Physica Heidel-berg Germany 1999
[7] B van Gasse G Deschrijver C Cornelis and E E KerreldquoFilters of residuated lattices and triangle algebrasrdquo InformationSciences vol 180 no 16 pp 3006ndash3020 2010
[8] E Turunen ldquoBoolean deductive systems of BL-algebrasrdquoArchive forMathematical Logic vol 40 no 6 pp 467ndash473 2001
[9] M Kondo ldquoFilters on commutative residuated latticesrdquo inIntegrated Uncertainty Management and Applications vol 68pp 343ndash347 Springer Berlin Germany 2010
[10] L Liu and K Li ldquoBoolean filters and positive implicative filtersof residuated latticesrdquo Information Sciences vol 177 no 24 pp5725ndash5738 2007
[11] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquoInformation Sciences vol 180 no 19 pp 3614ndash3632 2010
[12] MHaveshki A B Saeid and E Eslami ldquoSome types of filters in119861119871 algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006
[13] M Kondo andW A Dudek ldquoFilter theory of BL algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008
[14] A B Saeid and S Motamed ldquoNormal filters in BL-algebrasrdquoWorld Applied Sciences Journal vol 7 pp 70ndash76 2009
[15] E Turunen ldquoBL-algebras of basic fuzzy logicrdquoMathware amp SoftComputing vol 6 no 1 pp 49ndash61 1999
[16] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 no 3 pp 512ndash517 1971
[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
[18] J N Mordeson and D S Malik Fuzzy Commutative AlgebraWorld Scientific London UK 1998
[19] T Head ldquoA metatheorem for deriving fuzzy theorems fromcrisp versionsrdquo Fuzzy Sets and Systems vol 73 no 3 pp 349ndash358 1995
[20] T Head ldquoErratum to ldquoA metatheorem for deriving fuzzytheorems from crisp versionsrsquordquo Fuzzy Sets and Systems vol 79no 2 pp 277ndash278 1996
[21] I Jahan ldquoThe lattice of 119871-ideals of a ring is modularrdquo Fuzzy Setsand Systems vol 199 pp 121ndash129 2012
[22] Y B Jun Y Xu and X H Zhang ldquoFuzzy filters of MTL-algebrasrdquo Information Sciences vol 175 no 1-2 pp 120ndash1382005
[23] L Liu and K Li ldquoFuzzy filters of 119861119871-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005
[24] J L Zhang and H J Zhou ldquoFuzzy filters on the residuatedlatticesrdquo New Mathematics and Natural Compution vol 2 no1 pp 11ndash28 2006
[25] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[26] Z Xue Y XiaoW Liu H Cheng and Y Li ldquoIntuitionistic fuzzyfilter theory of 119861119871-algebrasrdquo International Journal of MachineLearning and Cybernetics vol 4 no 6 pp 659ndash669 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
ge ⋀
119910otimes119911le119909
119910 = 1119911 = 1
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
and [(V119862or V
119860)V119862orV119861
(1) or (V119862or V
119860) (119910 or 119909)]
and [(V119862or V
119861)V119862orV119860
(1) and (V119862or V
119861) (119911 or 119909)]
= ⋀
119910otimes119911le119909
(V119862or V
119860)V119862orV119861
(119910 or 119909)
or(V119862or V
119861)V119862orV119860
(119911 or 119909)
(18)
Let119910or119909 = 1199101015840 and 119911or119909 = 1199111015840 it is easy to verify that1199101015840otimes1199111015840 le 119909and then the above can be written as
= ⋀
1199101015840otimes1199111015840le119909
(V119862or V
119860)V119862orV119861
(1199101015840) or (V
119862or V
119861)V119862orV119860
(1199111015840)
= (V119862or V
119860)V119862orV119861
⊙(V119862or V
119861)V119862orV119860
(119909)
(19)
Thus 119906119862and 119906
119906119861
119860otimes119906
119906119860
119861le (119906
119862and 119906
119860)119906119862and119906119861 otimes(119906
119862and 119906
119861)119906119862and119906119860 and
V119862(119909) or VV119861
119860⊙VV119860
119861(119909) ge (V
119862or V
119860)V119862orV119861 ⊙(V
119862or V
119861)V119862orV119860 that is
the distributivity holds
4 Conclusions
In this paper by the notion of tip-extended pair of intuitionis-tic fuzzy sets it is verified that the set of all intuitionistic fuzzyfilters forms a bounded distributive lattice
Future research will focus on the lattice of fuzzy notionsin other algebras by tip-extended pair
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by AMEP of Linyi Universityand the Natural Science Foundation of Shandong Province(Grant no ZR2013FL006)
References
[1] M Ward and R P Dilworth ldquoResiduated latticesrdquo Transactionsof theAmericanMathematical Society vol 45 no 3 pp 335ndash3541939
[2] F Esteva and L Godo ldquoMonoidal 119905-norm based logic towards alogic for left-continuous t-normsrdquo Fuzzy Sets and Systems vol124 no 3 pp 271ndash288 2001
[3] P Hajek Metamathematics of Fuzzy Logic Kluwer AcademicDordrecht The Netherlands 1998
[4] C C Chang ldquoAlgebraic analysis of many valued logicsrdquo Trans-actions of the American Mathematical Society vol 88 pp 467ndash490 1958
[5] G J Wang An Introduction to Mathematical Logic and Resolu-tion Principle Science in China Press Beijing China 2003
[6] E Turunen Mathematics Behind Fuzzy Logic Physica Heidel-berg Germany 1999
[7] B van Gasse G Deschrijver C Cornelis and E E KerreldquoFilters of residuated lattices and triangle algebrasrdquo InformationSciences vol 180 no 16 pp 3006ndash3020 2010
[8] E Turunen ldquoBoolean deductive systems of BL-algebrasrdquoArchive forMathematical Logic vol 40 no 6 pp 467ndash473 2001
[9] M Kondo ldquoFilters on commutative residuated latticesrdquo inIntegrated Uncertainty Management and Applications vol 68pp 343ndash347 Springer Berlin Germany 2010
[10] L Liu and K Li ldquoBoolean filters and positive implicative filtersof residuated latticesrdquo Information Sciences vol 177 no 24 pp5725ndash5738 2007
[11] Y Zhu and Y Xu ldquoOn filter theory of residuated latticesrdquoInformation Sciences vol 180 no 19 pp 3614ndash3632 2010
[12] MHaveshki A B Saeid and E Eslami ldquoSome types of filters in119861119871 algebrasrdquo Soft Computing vol 10 no 8 pp 657ndash664 2006
[13] M Kondo andW A Dudek ldquoFilter theory of BL algebrasrdquo SoftComputing vol 12 no 5 pp 419ndash423 2008
[14] A B Saeid and S Motamed ldquoNormal filters in BL-algebrasrdquoWorld Applied Sciences Journal vol 7 pp 70ndash76 2009
[15] E Turunen ldquoBL-algebras of basic fuzzy logicrdquoMathware amp SoftComputing vol 6 no 1 pp 49ndash61 1999
[16] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 no 3 pp 512ndash517 1971
[17] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
[18] J N Mordeson and D S Malik Fuzzy Commutative AlgebraWorld Scientific London UK 1998
[19] T Head ldquoA metatheorem for deriving fuzzy theorems fromcrisp versionsrdquo Fuzzy Sets and Systems vol 73 no 3 pp 349ndash358 1995
[20] T Head ldquoErratum to ldquoA metatheorem for deriving fuzzytheorems from crisp versionsrsquordquo Fuzzy Sets and Systems vol 79no 2 pp 277ndash278 1996
[21] I Jahan ldquoThe lattice of 119871-ideals of a ring is modularrdquo Fuzzy Setsand Systems vol 199 pp 121ndash129 2012
[22] Y B Jun Y Xu and X H Zhang ldquoFuzzy filters of MTL-algebrasrdquo Information Sciences vol 175 no 1-2 pp 120ndash1382005
[23] L Liu and K Li ldquoFuzzy filters of 119861119871-algebrasrdquo InformationSciences vol 173 no 1ndash3 pp 141ndash154 2005
[24] J L Zhang and H J Zhou ldquoFuzzy filters on the residuatedlatticesrdquo New Mathematics and Natural Compution vol 2 no1 pp 11ndash28 2006
[25] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[26] Z Xue Y XiaoW Liu H Cheng and Y Li ldquoIntuitionistic fuzzyfilter theory of 119861119871-algebrasrdquo International Journal of MachineLearning and Cybernetics vol 4 no 6 pp 659ndash669 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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