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Research ArticleProduct of the Generalized πΏ-Subgroups
Dilek Bayrak and Sultan Yamak
Department of Mathematics, Namik Kemal University, 59000 Tekirdag, Turkey
Correspondence should be addressed to Dilek Bayrak; [email protected]
We introduce the notion of (π, π)-product of πΏ-subsets. We give a necessary and sufficient condition for (π, π)-πΏ-subgroup of aproduct of groups to be (π, π)-product of (π, π)-πΏ-subgroups.
1. Introduction
Starting from 1980, by the concept of quasi-coincidence ofa fuzzy point with a fuzzy set given by Pu and Liu [1], thegeneralized subalgebraic structures of algebraic structureshave been investigated again. Using the concept mentionedabove, Bhakat and Das [2, 3] gave the definition of (πΌ, π½)-fuzzy subgroup, where πΌ, π½ are any two of {β, π, ββ¨π, β β§π}with πΌ =ββ§π. The (β, ββ¨π)-fuzzy subgroup is an importantand useful generalization of fuzzy subgroups that were laidby Rosenfeld in [4]. After this, many other researchers usedthe idea of the generalized fuzzy sets that give several charac-terization results in different branches of algebra (see [5β10]).In recent years,many researchersmake generalizationswhichare referred to as (π, π)-fuzzy substructures and (βπ, βπβ¨ππ)-fuzzy substructures on this topic (see [11β15]).
Identifying the subgroups of a Cartesian product ofgroups plays an essential role in studying group theory. Manyimportant results on characterization of Cartesian product ofsubgroups, fuzzy subgroups, andππΏ-fuzzy subgroups exist inliterature. Chon obtained a necessary and sufficient conditionfor a fuzzy subgroup of a Cartesian product of groups tobe product of fuzzy subgroups under minimum operation[16]. Later, some necessary and sufficient conditions for aππΏ-subgroup of a Cartesian product of groups to be a π-product of ππΏ-subgroups were given by Yamak et al. [17]. Asubgroup of a Cartesian product of groups is characterizedby subgroups in the same study. Consequently, it seems to beinteresting to extend this study to generalized πΏ-subgroups.In this paper, we introduce the notion of the (π, π)-product
of πΏ-subsets and investigate some properties of the (π, π)-product of πΏ-subgroups. Also, we give a necessary andsufficient condition for (π, π)-πΏ-subgroup of a Cartesianproduct of groups to be a product of (π, π)-πΏ-subgroups.
2. Preliminaries
In this section, we start by giving some known definitions andnotations. Throughout this paper, unless otherwise stated, πΊalways stands for any given groupwith amultiplicative binaryoperation, an identity π and πΏ denote a complete lattice withtop and bottom elements 1, 0, respectively.
An πΏ-subset ofπ is any function fromπ into πΏ, which isintroduced by Goguen [18] as a generalization of the notionof Zadehβs fuzzy subset [19]. The class of πΏ-subsets of π willbe denoted by πΉ(π, πΏ). In particular, if πΏ = [0, 1], then it isappropriate to replace πΏ-subset with fuzzy subset. In this casethe set of all fuzzy subsets of π is denoted by πΉ(π). Let π΄and π΅ be πΏ-subsets of π. We say that π΄ is contained in π΅ ifπ΄(π₯) β€ π΅(π₯) for every π₯ β π and is denoted by π΄ β€ π΅. Thenβ€ is a partial ordering on the set πΉ(π, πΏ).
Definition 1 (see [20]). An πΏ-subset of πΊ is called an πΏ-subgroup of πΊ if, for all π₯, π¦ β πΊ, the following conditionshold:(G1) π΄(π₯) β§ π΄(π¦) β€ π΄(π₯π¦).(G2) π΄(π₯) β€ π΄(π₯β1).In particular, when πΏ = [0, 1], an πΏ-subgroup of πΊ is
referred to as a fuzzy subgroup of πΊ.
Hindawi Publishing CorporationJournal of MathematicsVolume 2016, Article ID 4918948, 5 pageshttp://dx.doi.org/10.1155/2016/4918948
2 Journal of Mathematics
Definition 2 (see [12]). Let π, π β πΏ and π < π. Let π΄ be anπΏ-subset of πΊ. π΄ is called (π, π)-πΏ-subgroup of πΊ if, for allπ₯, π¦ β πΊ, the following conditions hold:
(ii) π΄(π₯β1) β¨ π β₯ π΄(π₯) β§ π.
Denote by πΉπ(π, π, πΊ, πΏ) the set of all (π, π)-πΏ-subgroups ofπΊ. When πΏ = [0, 1], its counterpart is written as πΉπ(π, π, πΊ).
Unless otherwise stated, πΏ always represents any givendistributive lattice.
3. Product of (π,π)-πΏ-Subgroups
Definition 3 (see [16]). Let π΄ π be an πΏ-subset of πΊπ for eachπ = 1, 2, . . . , π. Then product of π΄ π (π = 1, 2, . . . , π) denotedby π΄1 Γ π΄2 Γ β β β Γ π΄π is defined to be the πΏ-subset of πΊ1 ΓπΊ2 Γ β β β Γ πΊπ that satisfies
Corollary 6. If π΄1, π΄2, . . . , π΄π are (π, π)-πΏ-subgroups ofπΊ1, πΊ2, . . . , πΊπ, respectively, then π΄1 Γ π΄2 Γ β β β Γ π΄π is (π, π)-πΏ-subgroup of πΊ1 Γ πΊ2 Γ β β β Γ πΊπ.
Theorem 7 (Theorem 2.9, [16]). Let πΊ1, πΊ2, . . . , πΊπ begroups, let π1, π2, . . . , ππ be identities, respectively, and letπ΄ be a fuzzy subgroup in πΊ1 Γ πΊ2 Γ β β β Γ πΊπ. Thenπ΄(π1, π2, . . . , ππβ1, π₯π, ππ+1, . . . , ππ) β₯ π΄(π₯1, π₯2, . . . , π₯π) for π =1, 2, . . . , πβ1, π+1, . . . , π if and only ifπ΄ = π΄1Γπ΄2Γβ β β Γπ΄π,where π΄1, π΄2, . . . , π΄π are fuzzy subgroups of πΊ1, πΊ2, . . . , πΊπ,respectively.
The following example shows that Theorem 7 may not betrue for any (π, π)-πΏ-subgroup.
Example 8. Consider
π΄ (π₯) =
{{{{{{{
{{{{{{{
{
0.8, π₯ = (0, 0) ,
0.7, π₯ = (1, 0) ,
0.6, π₯ = (0, 1) ,
0.5, π₯ = (1, 1) .
(6)
It is easy to see that π΄ is (0, 0.5)-fuzzy subgroup of Z2 Γ Z2.Since π΄(1, 0) β₯ π΄(1, 1) and π΄(0, 1) β₯ π΄(1, 1), π΄ satisfies the
Journal of Mathematics 3
condition of Theorem 7, but there do not exist π΄1, π΄2 fuzzysubgroups of Z2 Γ Z2 such that π΄ = π΄1 Γ π΄2.
In fact, suppose that there exist π΄1, π΄2 β πΉπ(0, 0.5,Z2 ΓZ2) such that π΄ = π΄1 Γπ΄2. Since π΄(1, 0) = 0.7 and π΄(0, 1) =0.6, we have π΄1(1) β₯ 0.7 and π΄2(1) β₯ 0.6. Hence
Example 10. We define the fuzzy subsets π΄ and π΅ of Z andZ2, respectively, as in Example 4. Then (0.4, 0.6)-product ofπ΄ and π΅ is as follows:
π΄Γ0.4
0.6π΅ (π₯) =
{
{
{
0.5, π₯ β 2Z Γ {0} ,
0.4, otherwise.(9)
Lemma 11. Let πΊ1, πΊ2, . . . , πΊπ be groups. Then we have thefollowing:
(1) If π΄ is (π, π)-πΏ-subgroup of πΊ1 Γ πΊ2Γ, . . . , πΊπ andπ΄ π(π₯) = π΄(π1, π2, . . . , ππβ1, π₯, ππ+1, . . . , ππ) for π =
1, 2, . . . , π, then π΄ π is (π, π)-πΏ-subgroup of πΊπ for allπ = 1, 2, . . . , π.
(2) If π΄ π is (π, π)-πΏ-subgroup of πΊπ for all π = 1, 2, . . . , π,then π΄1Γπππ΄2Γ
π
πβ β β Γπππ΄π is (π, π)-πΏ-subgroup of πΊ1 Γ
πΊ2Γ, . . . , πΊπ.
Proof. (1) Let π₯, π¦ β πΊπ. Since π΄ β πΉπ(π, π, πΊ1 Γ πΊ2Γ,
The following example shows that Corollary 15 may notbe true when π = 0.
Example 16. Consider
π΄ (π₯) =
{{{{{{{
{{{{{{{
{
0.4, π₯ = (0, 0) ,
0.3, π₯ = (1, 0) ,
0.2, π₯ = (0, 1) ,
0.1, π₯ = (1, 1) .
(18)
π΄ is (0.2, 0.5)-fuzzy subgroup of Z2 Γ Z2 and satisfies thenecessary condition of Corollary 15. But there is not any π΄1and π΄2, (0.2, 0.5)-fuzzy subgroup of Z2 Γ Z2, which holdπ΄ = π΄1Γ
0.2
0.5π΄2.
4. Conclusion
In this study, we give a necessary and sufficient condition for(0, π)-πΏ-subgroup of a Cartesian product of groups to be aproduct of (0, π)-πΏ-subgroups. The results obtained are notvalid for π = 0, and a counterexample is provided.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
[1] P.M. Pu and Y.M. Liu, βFuzzy topology. I. Neighborhood struc-ture of a fuzzy point and Moore-Smith convergence,β Journal ofMathematical Analysis and Applications, vol. 76, no. 2, pp. 571β599, 1980.
[2] S. K. Bhakat and P. Das, β(β, β β¨q)-fuzzy subgroup,β Fuzzy Setsand Systems, vol. 80, no. 3, pp. 359β368, 1996.
[3] S. K. Bhakat and P. Das, βOn the definition of a fuzzy subgroup,βFuzzy Sets and Systems, vol. 51, no. 2, pp. 235β241, 1992.
[4] A. Rosenfeld, βFuzzy groups,β Journal of Mathematical Analysisand Applications, vol. 35, pp. 512β517, 1971.
[5] M. Akram, K. H. Dar, and K. P. Shum, βInterval-valued (πΌ,π½)-fuzzy K-algebras,β Applied Soft Computing Journal, vol. 11, no. 1,pp. 1213β1222, 2011.
[6] M. Akram, B. Davvaz, and K. P. Shum, βGeneralized fuzzy LIEideals of LIE algebras,β Fuzzy Systems and Mathematics, vol. 24,no. 4, pp. 48β55, 2010.
[7] M. Akram, βGeneralized fuzzy Lie subalgebras,β Journal of Gen-eralized Lie Theory and Applications, vol. 2, no. 4, pp. 261β268,2008.
[8] S. K. Bhakat, β(β, ββ¨q)-fuzzy normal, quasinormal andmaximalsubgroups,β Fuzzy Sets and Systems, vol. 112, no. 2, pp. 299β312,2000.
[9] M. Shabir, Y. B. Jun, and Y. Nawaz, βCharacterizations of regularsemigroups by (πΌ,π½)-fuzzy ideals,β Computers & Mathematicswith Applications, vol. 59, no. 1, pp. 161β175, 2010.
[10] X. Yuan, C. Zhang, and Y. Ren, βGeneralized fuzzy groups andmany-valued implications,β Fuzzy Sets and Systems, vol. 138, no.1, pp. 205β211, 2003.
[11] S. Abdullah, M. Aslam, T. A. Khan, and M. Naeem, βA newtype of fuzzy normal subgroups and fuzzy cosets,β Journal ofIntelligent and Fuzzy Systems, vol. 25, no. 1, pp. 37β47, 2013.
[12] D. Bayrak and S. Yamak, βThe lattice of generalized normal L-subgroups,β Journal of Intelligent and Fuzzy Systems, vol. 27, no.3, pp. 1143β1152, 2014.
[13] Y. Feng and P. Corsini, β(π, π)-Fuzzy ideals of ordered semi-groups,β Annals of Fuzzy Mathematics and Informatics, vol. 4,no. 1, pp. 123β129, 2012.
[14] Y. B. Jun, M. S. Kang, and C. H. Park, βFuzzy subgroups basedon fuzzy points,β Communications of the Korean MathematicalSociety, vol. 26, no. 3, pp. 349β371, 2011.
[15] B. Yao, β(π, π)-Fuzzy normal subgroups, (π, π)-fuzzy quotientsubgroups,βThe Journal of Fuzzy Mathematics, vol. 13, no. 3, pp.695β705, 2005.
[16] I. Chon, βFuzzy subgroups as products,β Fuzzy Sets and Systems,vol. 141, no. 3, pp. 505β508, 2004.
[17] S. Yamak, O. KazancΔ±, and D. Bayrak, βA solution to an openproblem on the T-product of TL-fuzzy subgroups,β Fuzzy Setsand Systems, vol. 178, pp. 102β106, 2011.
[18] J. A. Goguen, βL-fuzzy sets,β Journal of Mathematical Analysisand Applications, vol. 18, pp. 145β174, 1967.
[19] L. A. Zadeh, βFuzzy sets,β Information and Computation, vol. 8,pp. 338β353, 1965.
[20] J. N. Mordeson and D. S. Malik, Fuzzy Commutative Algebra,World Scientific, 1998.