On Commutators of Fuzzy Multigroups

Fuzzy multigroup is an application of fuzzy multiset to group theory. Although, a lots have been done on the theory of fuzzy multigroups, some group’s theoretic notions could still be investigated in fuzzy multigroup context. Certainly, the idea of commutator is one of such group’s theoretic notions yet to be studied in the environment of fuzzy multigroups. Hence, the aim of this article is to establish the notion of commutator in fuzzy multigroup setting. A number of some related results are obtained and characterized. Among several results that are obtained, it is established that, if A and B are fuzzy submultigroups of a fuzzy multigroup C , then [ A, B ] ⊆ A ∪ B holds. Some homomorphic properties of commutator in fuzzy multigroup context are discussed. The notion of admissible fuzzy submultisets A and B of C ∈ FMG ( X ) under an operator domain D is explicated, and it is shown that ( A, B ) and [ A, B ] are D -admissible.


Introduction
Fuzzy set theory proposed by Zadeh [26] (although with heated disagreement as at then) has been widely studied with applications ranging from engineering and computer science to medical diagnosis and social behavior, etc. In a way of extending the application of fuzzy sets to group theory, Rosenfeld [22] proposed the idea of fuzzy groups as an application of fuzzy sets to group theory and some number of results were obtained. Numerous studies have been carried out on some group theoretic notions in fuzzy group setting (cf. [1,4,13,18,19,20]).
By synthesizing the concepts of fuzzy sets and multisets (cf. [14]), the idea of fuzzy multisets or fuzzy bags was proposed in [25] as a generalization of fuzzy sets defined by a count membership function where Q is the set of all crisp bags or multisets from the unit interval I = [0, 1] and . From [24], a fuzzy multiset can also be characterized by a high-order function. In particular, a fuzzy multiset A can be characterized by a function We denote the set of all fuzzy multisets by F M S(X).
Definition 2.2 ( [15]). Let A and B be fuzzy multisets of X. Then Note that and denote minimum and maximum operations.

Definition 2.3 ([23])
. Suppose X is a group. Then, a fuzzy multiset A of X is called a fuzzy multigroup of X if the following conditions are satisfied: By implication, a fuzzy multiset A over X is called a fuzzy multigroup of a group X if It follows immediately from the definition that, where e is the identity element of X. The second condition above is strictly We denote the set of all fuzzy multigroups of X by F M G(X).
Then, the product A • B is defined to be a fuzzy multiset of X as follows: otherwise.
are subgroups of X.
Proposition 2.8. Let A ∈ F M G(X). Then, the set A [α] defined by  Definition 2.11 ([9]). Let X and Y be groups and let f : X → Y be a homomorphism. Suppose A and B are fuzzy multigroups of X and Y , respectively. Then, f induces a homomorphism from A to B which satisfies , is a fuzzy multiset over Y defined by (ii) the inverse image of B under f , denoted by f −1 (B), is a fuzzy multiset over X defined by for every automorphism, θ of X. That is, θ(A) ⊆ A for every θ ∈ Aut(X).
It follows from [12] that, every characteristic fuzzy submultigroup of a fuzzy multigroup is normal.

Commutator of fuzzy multigroups
Recall that the commutator of two elements x and y of a group X is the element [x, y] = x −1 y −1 xy ∈ X. If H and K are subgroups of X, then the commutator subgroup or derived subgroup [H, K] of X is generated by {[x, y]|x ∈ H, y ∈ K}. Now, the idea of commutator in fuzzy multigroup context is introduced.
The commutator of A and B is a fuzzy multigroup [A, B] of X generated by (A, B). Now, we present some properties of commutator in fuzzy multigroup context as follow: , where e is the identity element in A.
Proposition 3.5. Let x, y, z ∈ X and A be a commutative fuzzy multigroup of X. Then P. A. Ejegwa and J. M. Agbetayo Proof. For x, y, z ∈ X, we have Similarly, Hence, the result. (ii) and similarly, we have The result follows.
Lemma 3.6. If x, y, z ∈ X and A is a commutative fuzzy multigroup of X, we have Proof. Straightforward from Proposition 3.5.
Proposition 3.7. Let x, y, z ∈ X, and A be a fuzzy multigroup of X. Then, we , where e is the identity of X.
Proof. For x, y, z ∈ X, we have Setting a = xzx −1 yx, b = yxy −1 zy and c = zyz −1 xz. By observation, b and c can be obtained by cyclic permutation of x, y, z. Furthermore, we see that the result follows. Proof. By a given hypothesis, as x and z commute. Conjugating by y gives as y and z commute. Repeating this argument j times, we have Theorem 3.9. Let A ∈ F M G(X) and x, y, z ∈ X. If y commutes with z in A and X is abelian, then CM A ([x, y, z]) = CM A ([x, z, y).
Proof. To start with, we have We observe that xy −1 x −1 y and xz −1 x −1 z lies in X, and thus commute. It follows that Since y and z commute, then This completes the proof.
Theorem 3.10. Let A ∈ F M G(X) and x, y, z ∈ X. If [x, y] commutes with both x and y, then Proof. By synthesizing Lemma 3.6, we have   Proof. We establish the proof by induction on n. Let x ∈ X. If x is not a commutator in X. Then

CM (A,A) = 0 and CM
Suppose x is a commutator in X. Then, for a, b ∈ X we have and so the result follows for k = 1. Since Thus, . Hence, the result follows for n = k. Since A∩B is a fuzzy multigroup of X, it suffices to proves that (A, B) ⊆ A∩B. Let x ∈ X. If x is not a commutator, then  Proof. Let x ∈ X. Since [A, B] is a fuzzy multigroup generated by (A, B), if we prove that (A, B) = (B, A) we are done. Assume x is not a commutator in X, then x −1 is not a commutator and consequently,    Proof. If x is not a commutator in X, then CM (A,A) (x) = 0. Suppose x is a commutator, then       ((A, B)).

This implies that
Proof. Since [A, B] is generated by (A, B), then f ( [A, B]) is a fuzzy multigroup of Y generated by f ((A, B)), which is a direct consequent of Theorem 2.12.  ((A, B)).

Conclusion
This work further the study of fuzzy multigroup theory parallel to crisp group theory. The ideas of commutator of fuzzy multigroups and commutator fuzzy submultigroups of fuzzy multigroups were proposed and characterized with a number of some related results. Some homomorphic images and preimages of commutator of fuzzy multigroups were considered with some results. The notion of admissible fuzzy submultisets A and B of C ∈ F M G(X) under an operator domain D was explicated, and it was shown that (A, B) and [A, B] are D-admissible. However, more properties of commutator in fuzzy multigroup setting could be exploited in future investigation.