Abstract
We introduce the notion of -product of -subsets. We give a necessary and sufficient condition for --subgroup of a product of groups to be -product of --subgroups.
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 -fuzzy subgroup, where , are any two of with . The -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 -fuzzy substructures and -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, and -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 -subgroup of a Cartesian product of groups to be a -product of -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 -subgroups. In this paper, we introduce the notion of the -product of -subsets and investigate some properties of the -product of -subgroups. Also, we give a necessary and sufficient condition for --subgroup of a Cartesian product of groups to be a product of --subgroups.
2. Preliminaries
In this section, we start by giving some known definitions and notations. Throughout this paper, unless otherwise stated, always stands for any given group with a multiplicative binary operation, an identity and denote a complete lattice with top and bottom elements 1, 0, respectively.
An -subset of is any function from into , which is introduced by Goguen [18] as a generalization of the notion of Zadeh’s fuzzy subset [19]. The class of -subsets of will be denoted by . In particular, if , then it is appropriate to replace -subset with fuzzy subset. In this case the set of all fuzzy subsets of is denoted by . Let and be -subsets of . We say that is contained in if for every and is denoted by . Then is a partial ordering on the set .
Definition 1 (see [20]). An -subset of is called an -subgroup of if, for all , the following conditions hold: (G1).(G2).In particular, when , an -subgroup of is referred to as a fuzzy subgroup of .
Definition 2 (see [12]). Let and . Let be an -subset of . is called --subgroup of if, for all , the following conditions hold: (i).(ii).Denote by the set of all --subgroups of . When , its counterpart is written as .
Unless otherwise stated, always represents any given distributive lattice.
3. Product of --Subgroups
Definition 3 (see [16]). Let be an -subset of for each . Then product of denoted by is defined to be the -subset of that satisfies
Example 4. We define the fuzzy subsets and of and , respectively, as follows: We obtain that
Theorem 5. Let be groups and such that is --subgroup of for each . Then is --subgroup of .
Proof. Let . ThenSimilarly, it can be shown that
Corollary 6. If are --subgroups of , respectively, then is --subgroup of .
Theorem 7 (Theorem , [16]). Let be groups, let be identities, respectively, and let be a fuzzy subgroup in . Then for if and only if , where are fuzzy subgroups of , respectively.
The following example shows that Theorem 7 may not be true for any --subgroup.
Example 8. ConsiderIt is easy to see that is -fuzzy subgroup of . Since and , satisfies the condition of Theorem 7, but there do not exist , fuzzy subgroups of such that .
In fact, suppose that there exist such that . Since and , we have and . Hencea contradiction.
Definition 9. Let be an -subset of for each . Then -product of denoted by is defined to be an -subset of that satisfies
Example 10. We define the fuzzy subsets and of and , respectively, as in Example 4. Then -product of and is as follows:
Lemma 11. Let be groups. Then we have the following: (1)If is --subgroup of and for , then is --subgroup of for all .(2)If is --subgroup of for all , then is --subgroup of .
Proof. (1) Let . Since , we haveSimilarly, we can show that . Hence, by Definition 2 for .
(2) Let . Then,Similarly, we can show thatHence, is --subgroup of .
Theorem 12. Let be groups and . Then for if and only if , where .
Proof. Suppose that for . Now, for any , we haveHence we obtain that . Conversely, assume that . Then, we obtainOn the other hand, .
Since ,
Hence,
Similarly, we get for .
Lemma 13. Let be groups, let be identities of , respectively, and . If for , then
Proof. By Definition 2, we observe that
Lemma 14. Let be groups and . Then for if and only if for .
Proof. Now assume that for . Next, for any , we haveConversely, assume that for . By Definition 2 and Lemma 11, we obtain thatHence, . Similarly, we get for .
As a consequence of Theorem 12 and Lemma 14, we have the following corollary.
Corollary 15. Let be groups and . Then for if and only if , where .
The following example shows that Corollary 15 may not be true when .
Example 16. Consider is -fuzzy subgroup of and satisfies the necessary condition of Corollary 15. But there is not any and , -fuzzy subgroup of , which hold .
4. Conclusion
In this study, we give a necessary and sufficient condition for --subgroup of a Cartesian product of groups to be a product of --subgroups. The results obtained are not valid for , and a counterexample is provided.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.