# Product of the GeneralizedL-Subgroups

Authors: Dilek Bayrak; Sultan Yamak
Journal: Journal of Mathematics
 (2016)

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## Abstract

We introduce the notion of(λ,μ)-product of L-subsets. We give a necessary and sufficient condition for (λ,μ)-L-subgroup of a product of groups to be (λ,μ)-product of (λ,μ)-L-subgroups.


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## Body

## 1. Introduction

Starting from 1980, by the concept of quasi-coincidence of a fuzzy point with a fuzzy set given by Pu and Liu [1], the generalized subalgebraic structures of algebraic structures have been investigated again. Using the concept mentioned above, Bhakat and Das [2, 3] gave the definition of (α,β)-fuzzy subgroup, where α,  β are any two of {∈,q,∈∨q,∈∧q} with α≠∈∧q. The (∈,∈∨q)-fuzzy subgroup is an important and useful generalization of fuzzy subgroups that were laid by Rosenfeld in [4]. After this, many other researchers used the idea of the generalized fuzzy sets that give several characterization results in different branches of algebra (see [5–10]). In recent years, many researchers make generalizations which are referred to as (λ,μ)-fuzzy substructures and (∈λ,∈λ∨qμ)-fuzzy substructures on this topic (see [11–15]).Identifying the subgroups of a Cartesian product of groups plays an essential role in studying group theory. Many important results on characterization of Cartesian product of subgroups, fuzzy subgroups, andTL-fuzzy subgroups exist in literature. Chon obtained a necessary and sufficient condition for a fuzzy subgroup of a Cartesian product of groups to be product of fuzzy subgroups under minimum operation [16]. Later, some necessary and sufficient conditions for a TL-subgroup of a Cartesian product of groups to be a T-product of TL-subgroups were given by Yamak et al. [17]. A subgroup of a Cartesian product of groups is characterized by subgroups in the same study. Consequently, it seems to be interesting to extend this study to generalized L-subgroups. In this paper, we introduce the notion of the (λ,μ)-product of L-subsets and investigate some properties of the (λ,μ)-product of L-subgroups. Also, we give a necessary and sufficient condition for (λ,μ)-L-subgroup of a Cartesian product of groups to be a product of (λ,μ)-L-subgroups.


## 2. Preliminaries

In this section, we start by giving some known definitions and notations. Throughout this paper, unless otherwise stated,G always stands for any given group with a multiplicative binary operation, an identity e and L denote a complete lattice with top and bottom elements 1, 0, respectively.AnL-subset of X is any function from X into L, which is introduced by Goguen [18] as a generalization of the notion of Zadeh’s fuzzy subset [19]. The class of L-subsets of X will be denoted by F(X,L). In particular, if L=[0,1], then it is appropriate to replace L-subset with fuzzy subset. In this case the set of all fuzzy subsets of X is denoted by F(X). Let A and B be L-subsets of X. We say that A is contained in B if A(x)≤B(x) for every x∈X and is denoted by A≤B. Then ≤ is a partial ordering on the set F(X,L).Definition 1 (see [20]).
AnL-subset of G is called an L-subgroup of G if, for all x,y∈G, the following conditions hold: (G1)
A(x)∧A(y)≤A(xy).(G2)
A(x)≤A(x-1).
In particular, whenL=[0,1], an L-subgroup of G is referred to as a fuzzy subgroup of G.Definition 2 (see [12]).
Letλ,μ∈L and λ<μ. Let A be an L-subset of G. A is called (λ,μ)-L-subgroup of G if, for all x,y∈G, the following conditions hold: (i)
A(xy)∨λ≥A(x)∧A(y)∧μ.(ii)
A(x-1)∨λ≥A(x)∧μ.Denote by FS(λ,μ,G,L) the set of all (λ,μ)-L-subgroups of G. When L=[0,1], its counterpart is written as FS(λ,μ,G).
Unless otherwise stated,L always represents any given distributive lattice.


## 3. Product of(λ,μ)-L-Subgroups

Definition 3 (see [16]).
LetAi be an L-subset of Gi for each i=1,2,…,n. Then product of Ai(i=1,2,…,n) denoted by A1×A2×⋯×An is defined to be the L-subset of G1×G2×⋯×Gn that satisfies(1)A1×A2×⋯×Anx1,x2,…,xn=A1x1∧A2x2∧⋯∧Anxn.Example 4.
We define the fuzzy subsetsA and B of Z and Z2, respectively, as follows: (2)Ax=0.5,x∈2Z,0.3,otherwise,Bx=0.7,x=0¯,0.2,x=1¯.We obtain that(3)A×Bx=0.5,x∈2Z×0¯,0.3,x∈Z-2Z×0¯,0.2,otherwise.Theorem 5.
LetG1,G2,…,Gn be groups and ⋁i=1nλi<⋀i=1nμi such that Ai is (λi,μi)-L-subgroup of Gi for each i=1,2,…,n. Then A1×A2×⋯×An is (⋁i=1nλi,⋀i=1nμi)-L-subgroup of G1×G2×⋯×Gn.Proof.
Let(x1,x2,…,xn),(y1,y2,…,yn)∈G1×G2×⋯×Gn. Then(4)A1×A2×⋯×Anx1,x2,…,xn·y1,y2,…,yn∨⋁i=1nλi=A1×A2×⋯×Anx1y1,x2y2,…,xnyn∨λ1∨⋁i=1nλi=A1x1y1∧A2x2y2∧⋯∧Anxnyn∨⋁i=1nλi≥A1x1y1∨λ1∧A2x2y2∨λ2∧⋯∧Anxnyn∨λn≥A1x1∧A1y1∧μ1∧A2x2∧A2y2∧μ2∧⋯∧Anxn∧Anyn∧μn=A1×A2×⋯×Anx1,x2,…,xn∧A1×A2×⋯×Any1,y2,…,yn∧⋀i=1nμi.Similarly, it can be shown that(5)A1×A2×⋯×Anx1,x2,…,xn-1∨⋁i=1nλi≥A1×A2×⋯×Anx1,x2,…,xn∧⋀i=1nμi.Corollary 6.
IfA1,A2,…,An are (λ,μ)-L-subgroups of G1,G2,…,Gn, respectively, then A1×A2×⋯×An is (λ,μ)-L-subgroup of G1×G2×⋯×Gn.Theorem 7 (Theorem2.9, [16]).
LetG1,G2,…,Gn be groups, let e1,e2,…,en be identities, respectively, and let A be a fuzzy subgroup in G1×G2×⋯×Gn. Then A(e1,e2,…,ei-1,xi,ei+1,…,en)≥A(x1,x2,…,xn) for i=1,2,…,k-1,k+1,…,n if and only if A=A1×A2×⋯×An, where A1,A2,…,An are fuzzy subgroups of G1,G2,…,Gn, respectively.The following example shows that Theorem7 may not be true for any (λ,μ)-L-subgroup.Example 8.
Consider(6)Ax=0.8,x=0¯,0¯,0.7,x=1¯,0¯,0.6,x=0¯,1¯,0.5,x=1¯,1¯.It is easy to see that A is (0,0.5)-fuzzy subgroup of Z2×Z2. Since A(1¯,0¯)≥A(1¯,1¯) and A(0¯,1¯)≥A(1¯,1¯), A satisfies the condition of Theorem 7, but there do not exist A1,  A2 fuzzy subgroups of Z2×Z2 such that A=A1×A2.
In fact, suppose that there existA1,A2∈FS(0,0.5,Z2×Z2) such that A=A1×A2. Since A(1¯,0¯)=0.7 and A(0¯,1¯)=0.6, we have A1(1¯)≥0.7 and A2(1¯)≥0.6. Hence(7)0.5=A1×A21¯,1¯=A11¯∧A21¯≥0.7∧0.6=0.6,a contradiction.Definition 9.
LetAi be an L-subset of Gi for each i=1,2,…,n. Then (λ,μ)-product of Ai(i=1,2,…,n) denoted by A1×μλA2×μλ⋯×μλAn is defined to be an L-subset of G1×G2×⋯×Gn that satisfies(8)A1×μλA2×μλ⋯×μλAnx1,x2,…,xn=A1x1∧A2x2∧⋯∧Anxn∧μ∨λ=⋀i=1nAixi∧μ∨λ.Example 10.
We define the fuzzy subsetsA and B of Z and Z2, respectively, as in Example 4. Then (0.4,0.6)-product of A and B is as follows:(9)A×0.60.4Bx=0.5,x∈2Z×0¯,0.4,otherwise.Lemma 11.
LetG1,G2,…,Gn be groups. Then we have the following: (1)
IfA is (λ,μ)-L-subgroup of G1×G2×,…,Gn and Ai(x)=A(e1,e2,…,ei-1,x,ei+1,…,en) for i=1,2,…,n, then Ai is (λ,μ)-L-subgroup of Gi for all i=1,2,…,n.(2)
IfAi is (λ,μ)-L-subgroup of Gi for all i=1,2,…,n, then A1×μλA2×μλ⋯×μλAn is (λ,μ)-L-subgroup of G1×G2×,…,Gn.Proof.
(1) Letx,y∈Gi. Since A∈FS(λ,μ,G1×G2×,…,Gn,L), we have(10)Aix∧Aiy∧μ=Ae1,e2,…,ei-1,x,ei+1,…,en∧Ae1,e2,…,ei-1,y,ei+1,…,en∧μ≤Ae1,e2,…,ei-1,x·y,ei+1,…,en∨λ=Aix·y∨λ.Similarly, we can show that Ai(x)∧μ≤Ai(x-1)∨λ. Hence, Ai∈FS(λ,μ,Gi,L) by Definition 2 for i=1,2,…,n.
(2) Let(x1,x2,…,xn),(y1,y2,…,yn)∈G1×G2×,…,Gn. Then,(11)A1×μλA2×μλ⋯×μλAnx1,x2,…,xny1,y2,…,yn∨λ=A1×μλA2×μλ⋯×μλAnx1y1,x2y2,…,xnyn∨λ=⋀i=1nAixiyi∧μ∨λ=A1x1y1∨λ∧A2x2y2∨λ∧⋯∧Anxnyn∨λ∧μ∨λ≥A1x1∧A1y1∧μ∧A2x2∧A2y2∧μ∧⋯∧Anxn∧Anyn∧μ∨λ=⋀i=1nAixi∧μ∨λ∧⋀i=1nAiyi∧μ∨λ∧μ=A1×μλA2×μλ⋯×μλAnx1,x2,…,xn∧A1×μλA2×μλ⋯×μλAny1,y2,…,yn∧μ.Similarly, we can show that(12)A1×μλA2×μλ⋯×μλAnx1,x2,…,xn-1∨λ≥A1×μλA2×μλ⋯×μλAnx1,x2,…,xn∧μ.Hence, A1×μλA2×μλ⋯×μλAn is (λ,μ)-L-subgroup of G1×G2×,…,Gn.Theorem 12.
LetG1,G2,…,Gn be groups and A∈FS(λ,μ,G1×G2×⋯×Gn,L). Then (A(e1,e2,…,ei-1,xi,ei+1,…,en) ∧A(x1,x2,…,xi-1,ei,xi+1,…,xn)∧μ) ∨λ=A(x1,x2,…,xn) for i=1,2,…,n if and only if A=A1×μλA2×μλ⋯×μλAn, where Ai(x)=A(e1,e2,…,ei-1,x,ei+1,…,en).Proof.
Suppose that(A(e1,e2,…,ei-1,xi,ei+1,…,en)∧ A(x1,x2,…,xi-1,ei,xi+1,…,xn)∧μ)∨λ= Ax1,x2,…,xn for i=1,2,…,n. Now, for any x1,x2,…,xn∈G1×G2×⋯×Gn, we have(13)Ax1,x2,…,xn=Ax1,e2,…,en∧Ae1,x2,…,xn∧μ∨λ=A1x1∧Ae1,x2,…,en∧Ae1,e2,x3,…,xn∧μ∨λ∧μ∨λ=A1x1∧A2x2∧Ae1,e2,x3,…,xn∧μ∨λ⋮=A1x1∧A2x2∧⋯∧Ae1,e2,…,xn∧μ∨λ=A1×μλA2×μλ⋯×μλAnx1,x2,…,xn.Hence we obtain that A=A1×μλA2×μλ⋯×μλAn. Conversely, assume that A=A1×μλA2×μλ⋯×μλAn. Then, we obtain(14)Ax1,x2,…,xn=A1×μλA2×μλ⋯×μλAnx1,x2,…,xn=Ax1,e2,…,en∧Ae1,x2,…,en∧⋯∧Ae1,e2,…,xn∧μ∨λ≤Ax1,e2,…,en∧Ae1,x2,…,en∧⋯∧Ae1,e2,…,xn-1,xn∨λ∧μ∨λ=Ax1,e2,…,en∧Ae1,x2,…,en∧⋯∧Ae1,e2,…,xn-1,xn∧μ∨λ⋮≤Ax1,e2,…,en∧Ae1,x2,…,xn∧μ∨λ.On the other hand, Ax1,e2,…,en∧Ae1,x2,…,xn∧μ∨λ≤Ax1,x2,…,xn∨λ.
SinceA(x1,x2,…,xn)≥λ,  Ax1,e2,…,en∧Ae1,x2,…,xn∧μ∨ λ≤Ax1,x2,…,xn.
Hence,Ax1,x2,…,xn= Ax1,e2,…,en∧Ae1,x2,…,xn∧μ∨λ.
Similarly, we get(A(e1,e2,…,ei-1,xi,ei+1,…,en)∧ A(x1,x2,…,xi-1,ei,xi+1,…,xn)∧μ)∨ λ=Ax1,x2,…,xn for i=2,3,…,n.Lemma 13.
LetG1,G2,…,Gn be groups, let e1,e2,…,en be identities of G1,G2,…,Gn, respectively, and A∈FS(λ,μ,G1×G2×⋯×Gn). If (A(e1,e2,…,ei-1,xi,ei+1,…,en)∧μ)∨ λ≥A(x1,x2,…,xn) for i=1,2,…,k-1,k+1,…,n, then Ae1,e2,…,ek-1,xk,ek+1,…,en∧μ∨ λ≥Ax1,x2,…,xn.Proof.
By Definition2, we observe that(15)Ae1,e2,…,ek-1,xk,ek+1,…,en∧μ∨λ=Ax1,x2,…,xnx1-1,x2-1,…,xk-1-1,ek,xk+1-1,…,xn-1∧μ∨λ=Ax1,x2,…,xnx1-1,x2-1,…,xk-1-1,ek,xk+1-1,…,xn-1∨λ∧μ≥Ax1,x2,…,xn∧Ax1-1,x2-1,…,xk-1-1,ek,xk+1-1,…,xn-1∧μ∨λ=Ax1,x2,…,xn∨λ∧Ax1-1,x2-1,…,xk-1-1,ek,xk+1-1,…,xn-1∨λ∧μ≥Ax1,x2,…,xn∨λ∧Ax1,x2,…,xk-1,ek,xk+1,…,xn∧μ∨λ=Ax1,x2,…,xn∨λ∧Ax1,x2,…,xk-1,ek,xk+1,…,xn∨λ∧μ≥Ax1,x2,…,xn∨λ∧Ae1,…,ek-2,xk-1,ek,…,en∧Ax1,…,ek-1,ek,xk+1,…,xn∧μ∨λ=Ax1,x2,…,xn∨λ∧Ax1,…,ek-1,ek,xk+1,…,xn∧μ∨λ⋮≥Ax1,x2,…,xn∨λ≥Ax1,x2,…,xn.Lemma 14.
LetG1,G2,…,Gn be groups and A∈FS(0,μ,G1×G2×⋯×Gn,L). Then A(e1,e2,…,ei-1,xi,ei+1,…,en)∧A(x1,x2,…,xi-1,ei,xi+1,…,xn)∧μ=A(x1,x2,…,xn) for i=1,2,…,n if and only if A(e1,e2,…,ei-1,xi,ei+1,…,en)∧μ≥A(x1,x2,…,xn) for i=1,2,…,k-1,k+1,…,n.Proof.
Now assume thatA(e1,e2,…,ei-1,xi,ei+1,…,en)∧ A(x1,x2,…,xi-1,ei,xi+1,…,xn)∧ μ=Ax1,x2,…,xn for i=1,2,…,n. Next, for any x1,x2,…,xn∈G1×G2×⋯×Gn, we have(16)Ax1,x2,…,xn=Ae1,e2,…,ei-1,xi,ei+1,…,en∧Ax1,x2,…,xi-1,ei,xi+1,…,xn∧μ≤Ae1,e2,…,ei-1,xi,ei+1,…,en∧μ.Conversely, assume that A(e1,e2,…,ei-1,xi,ei+1,…,en)∧ μ≥A(x1,x2,…,xn) for i=1,2,…,k-1,k+1,…,n. By Definition 2 and Lemma 11, we obtain that(17)Ax1,x2,…,xn=Ax1,x2,…,xn∧Ax1,x2,…,xn∧⋯∧Ax1,x2,…,xn≤Ax1,e2,…,en∧Ae1,x2,…,en∧⋯∧Ae1,e2,…,xn∧μ≤Ax1,e2,…,en∧Ae1,x2,…,en∧⋯∧Ae1,e2,…,xn-1,xn∧μ⋮≤Ax1,e2,…,en∧Ae1,x2,…,xn∧μ≤Ax1,x2,…,xn.Hence, A(x1,x2,…,xn)=A(x1,e2,…,en)∧ A(e1,x2,…,xn)∧μ. Similarly, we get A(e1,e2,…,ei-1,xi,ei+1,…,en)∧ A(x1,x2,…,xi-1,ei,xi+1,…,xn)=A(x1,x2,…,xn) for i=2,3,…,n.As a consequence of Theorem12 and Lemma 14, we have the following corollary.Corollary 15.
LetG1,G2,…,Gn be groups and A∈FS(0,μ,G1×G2×⋯×Gn,L). Then A(e1,e2,…,ei-1,xi,ei+1,…,en)∧μ≥ A(x1,x2,…,xn) for i=1,2,…,k-1,k+1,…,n if and only if A=A1×μA2×μ⋯×μAn, where Ai(x)=A(e1,e2,…,ei-1,x,ei+1,…,en).The following example shows that Corollary15 may not be true when λ≠0.Example 16.
Consider(18)Ax=0.4,x=0¯,0¯,0.3,x=1¯,0¯,0.2,x=0¯,1¯,0.1,x=1¯,1¯.A is (0.2,0.5)-fuzzy subgroup of Z2×Z2 and satisfies the necessary condition of Corollary 15. But there is not any A1 and A2, (0.2,0.5)-fuzzy subgroup of Z2×Z2, which hold A=A1×0.50.2A2.


## 4. Conclusion

In this study, we give a necessary and sufficient condition for(0,μ)-L-subgroup of a Cartesian product of groups to be a product of (0,μ)-L-subgroups. The results obtained are not valid for λ≠0, and a counterexample is provided.



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