Concept of Anti Multigroups and its Properties

The concept of multigroups is an application of multiset to group theory. Multigroup is an algebraic structure of a multiset whose underlying set is a group. The objective of this paper is to introduce the concept of anti multigroups and deduce some related results. We establish that a multiset defined over a group is a multigroup if and only if its complement is an anti multigroup. Finally, some results that connect cuts of multigroups to anti multigroups are considered.


Introduction
The term multisets as buttressed by Knuth [22], was first suggested by N. G. de Bruijn (cf. [6]) in a private communication to D. E. Knuth, as an important generalization of set theory, by relaxing the idea of distinct collection of elements in a set. Multiset theory has been explored in literature [9,21,25,27]. The notion of multisets is a boost to the concept of multigroups via multisets, which generalizes group theory. Nazmul et al. [23] proposed the concept of multigroups in multisets framework and presented a number of results. The notion is parallel to fuzzy groups [24]. A comprehensive account on the concept of multigroups was carried out in [18], and it was established that multigroup via multiset is a generalization of group theory.
The concept of multigroups via multisets has been researched upon since inception.
denoted the number of times an object x occur in A. Whenever Any ordinary set B is actually a multiset , , B B χ where B χ is its characteristic function. The set X is called the ground or generic set of the class of all multisets containing objects from X.
Take X to be the set from which multisets are constructed. The multiset n X is the set of all multisets of X such that no element occurs more than n times. Likewise, the multiset ∞ X is the set of all multisets of X such that there is no limit on the number of , , Other forms of multiset representations can be found in literature.
Definition 2.2. [21] Let X be a nonempty set and n X be the multiset space defined over X. Then, for any implies the set of all multisets over X drawn from the multiset space . , where ∧ and ∨ denote minimum and maximum, respectively.
Definition 2.7. [15,23] Let X be a group. A multiset A of X is said to be a multigroup of X if it satisfies the following two conditions: The set of all multigroups of X is denoted by ( ).

X MG
It can be easily verified that if A is a multigroup of X, then is the tip of A, where e is the identity element of X.
Remark 2.1. [23] Let X be a group and A be a multiset over X. If then A is called a multigroup of X.
are called the strong and weak upper cuts of A. Cleary, ( ) [ ] .

Anti Multigroups and Some Properties
This section presents anti multigroup as a multigroup in reverse order. We denote a group by X unless otherwise stated.

Concept of anti multigroups
Here, we define anti multigroup and discuss some of its properties.
We denote the set of all anti multigroups of X by ( ).

X AMG
be a group such that . , , , where e is the identity element of X.
We present the verifications of (i) to (iii) as below.
(i) By Definition 3.2, This completes the proof of (i).

Proposition 3.2. If A and B are anti multigroups of X, then B A I
is an anti Proof. Let The proof is completed.

Theorem 3.2. If A and B are anti multigroups of X, then the sum of A and B is an anti multigroup of X.
Proof. Let . ,

Proposition 3.3. A multiset A is an anti multigroup of X if and only if
Assume that A is an anti multigroup of X. Then the following conditions hold; Conversely, suppose the given condition is satisfied. Combining the following facts: we conclude that A is an anti multigroup of X.

Proof. Let
, Since X is finite, x has a finite order. Thus Now using the definition of an anti multigroupoid repeatedly, it follows that Hence the result.
The result follows. where e is the identity of X.
Hence the result follows.

Cuts of anti multigroups
In this subsection, we propose the idea of cuts of anti multigroups and outline some results.

Conclusion
We have proposed the concept of anti multigroups and deduced some properties of anti multigroups. It was established that a multiset of a group is a multigroup if and only if the complement of the multiset is an anti multigroup. For future research, some analogous results in multigroups could be investigated in anti multigroup setting.