Recently, fuzzy multi-sets have come to the forefront of scientists’ interest and have been used in algebraic structures such as multi-groups, multi-rings, anti-fuzzy multigroup and (α, γ)-anti-fuzzy subgroups. In this paper, we first summarize the knowledge about the algebraic structure of fuzzy multi-sets such as (α, γ)-anti-multi-fuzzy subgroups. In a way, the notion of anti-fuzzy multigroup is an application of anti-fuzzy multi sets to the theory of group. The concept of anti-fuzzy multigroup is a complement of an algebraic structure of a fuzzy multi set that generalizes both the theories of classical group and fuzzy group. The aim of this paper is to highlight the connection between fuzzy multi-sets and algebraic structures from an anti-fuzzification point of view. Therefore, in this paper, we define (α, γ)-anti-multi-fuzzy subgroups, (α, γ)-anti-multi-fuzzy normal subgroups, (α, γ)-anti-multi-fuzzy homomorphism on (α, γ)-anti-multi-fuzzy subgroups and these been explicated some algebraic structures. Then, we introduce the concept (α, γ)-anti-multi-fuzzy subgroups and (α, γ)-anti-multi-fuzzy normal subgroups and of their properties. This new concept of homomorphism as a bridge among set theory, fuzzy set theory, anti-fuzzy multi sets theory and group theory and also shows the effect of anti-fuzzy multi sets on a group structure. Certain results that discuss the (α, γ) cuts of anti-fuzzy multigroup are explored.

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The outlines are presented as follows: Section 2 presents some foundational notions relevant to the study, whereas the main results are reported in Section 3. In Section 4, we make some concluding remarks and suggestions for future work.

In this paper,

The value

A fuzzy set on a set

Conversely, assume

So

In this way

If there exists

Then

Holds

Let

Note that

Hence

Therefore

This is a contradict with that

Assume that

Thus

Thus

If

If

If

If

If

(iii) If

(iv)

(v) Assume that

Since

Conversely, if

Therefore

The aim of this paper was to highlight the function between (α, γ)-anti-multi-fuzzy subgroups and algebraic structures from other a point of view. It is well known that the concept of fuzzy multi set is well established in dealing with many real-life problems. So, the algebraic structure defined concerning them in this paper would help to approach these problems with a different perspective.

In this paper, we have defined the notion of (α, γ)-anti-multi-fuzzy subgroups and this structure some algebraic properties were developed. In this article, we have discussed (α, γ)-anti-multi-fuzzy subgroups, (α, γ)-anti-multi-fuzzy normal subgroups and defined (α, γ)-anti-multi-fuzzy homomorphism on (α, γ)-anti-multi-fuzzy subgroups. Interestingly, it has been observed that (α, γ)-anti-multi-fuzzy concept adds another dimension to the defined anti-fuzzy multi normal subgroups. This concept can further be extended for new results.