In this section, a brief survey of the notion of msets as introduced by Yager [2], Blizard [1, 3], and Jena et al. [4] have been collected. Furthermore, the different types of collections of msets, the basic definitions, and notions of relations and functions in mset context are introduced by Girish and John [5–8]. Other important research about multiset theory and its applications can be found in [9–16].

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**Definition 1**

A collection of elements containing duplicates is called an mset. Formally, if *X* is a set of elements, an mset *M* drawn from the set *X* is represented by a function count *M* or *C*_{M} defined as \(C_{M}: X\rightarrow \mathbb {N}\) where \(\mathbb {N}\) represents the set of nonnegative integers.

Let *M* be an mset from the set *X*={*x*_{1},*x*_{2},…,*x*_{n}} with *x* appearing *n* times in *M*. It is denoted by *x*∈^{n}*M*. The mset *M* drawn from the set *X* is denoted by *M*={*k*_{1}/*x*_{1},*k*_{2}/*x*_{2},…,*k*_{n}/*x*_{n}} where *M* is an mset with *x*_{1} appearing *k*_{1} times, *x*_{2} appearing *k*_{2} times, and so on. In Definition 10, *C*_{M}(*x*) is the number of occurrences of the element *x* in the mset *M*. However, those elements which are not included in the mset *M* have zero count. An mset *M* is a set if *C*_{M}(*x*)=0 or 1 ∀ *x*∈*X*.

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**Definition 2**

A domain *X* is defined as a set of elements from which msets are constructed. The mset space [ *X*]^{m} is the set of all msets whose elements are in *X* such that no element in the mset occurs more than *m* times. The set [ *X*]^{∞} is the set of all msets over a domain *X* such that there is no limit on the number of occurrences of an element in an mset.

Let *M*,*N*∈[*X*]^{m}. Then, the following are defined:

- (1)
*M* is a submset of *N* denoted by (*M*⊆*N*) if *C*_{M}(*x*)≤*C*_{N}(*x*) ∀ *x*∈*X*.

- (2)
*M*=*N* if *M*⊆*N* and *N*⊆*M*.

- (3)
*M* is a proper submset of *N* denoted by (*M*⊂*N*) if *C*_{M}(*x*)≤*C*_{N}(*x*) ∀ *x*∈*X* and there exists at least one element *x*∈*X* such that *C*_{M}(*x*)<*C*_{N}(*x*).

- (4)
*P*=*M*∪*N* if *C*_{P}(*x*)= max{*C*_{M}(*x*),*C*_{N}(*x*)} for all *x*∈*X*.

- (5)
*P*=*M*∩*N* if *C*_{P}(*x*)= min{*C*_{M}(*x*),*C*_{N}(*x*)} for all *x*∈*X*.

- (6)
Addition of *M* and *N* results is a new mset *P*=*M*⊕*N* such that *C*_{P}(*x*)= min{*C*_{M}(*x*)+*C*_{N}(*x*),*m*} for all *x*∈*X*.

- (7)
Subtraction of *M* and *N* results in a new mset *P*=*M*⊖*N* such that *C*_{P}(*x*)= max{*C*_{M}(*x*)−*C*_{N}(*x*),0} for all *x*∈*X*, where ⊕ and ⊖ represent mset addition and mset subtraction, respectively.

- (8)
An mset *M* is empty if *C*_{M}(*x*)=0 ∀ *x*∈*X*.

- (9)
The support set of *M* denoted by *M*^{∗} is a subset of *X* and *M*^{∗}={*x*∈*X*:*C*_{M}(*x*)>0}; that is, *M*^{∗} is an ordinary set and it is also called root set.

- (10)
The cardinality of an mset *M* drawn from a set *X* is Card\((M)=\sum \limits _{x\in X} C_{M}(x)\).

- (11)
*M* and *N* are said to be equivalent if and only if Card (*M*)=*Card*(*N*).

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**Definition 3**

Let *M*∈[ *X*]^{m} and *N*⊆*M*. Then, the complement *N*^{c} of *N* in [ *X*]^{m} is an element of [ *X*]^{m} such that *N*^{c}=*M*⊖*N*.

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**Definition 4**

A submset *N* of *M* is a whole submset of *M* with each element in *N* having full multiplicity as in *M*; that is, *C*_{N}(*x*)=*C*_{M}(*x*) for every *x*∈*N*^{∗}.

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**Definition 5**

A submset *N* of *M* is a partial whole submset of *M* with at least one element in *N* having full multiplicity as in *M*. i.e., *C*_{N}(*x*)=*C*_{M}(*x*) for some *x*∈*N*^{∗}.

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**Definition 6**

A submset *N* of *M* is a full submset of *M* if each element in *M* is an element in *N* with the same or lesser non-zero multiplicity as in *M*, i.e., *M*^{∗}=*N*^{∗} with *C*_{N}(*x*)≤*C*_{M}(*x*) for every *x*∈*N*^{∗}.

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**Definition 7**

Let *M*∈[*X*]^{m}. The power whole mset of *M* denoted by *P**W*(*M*) is defined as the set of all whole submsets of *M*.

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**Definition 8**

Let *M*∈[ *X*]^{m}. The power full msets of *M*, *P**F*(*M*), is defined as the set of all full submsets of *M*. The cardinality of *P**F*(*M*) is the product of the counts of the elements in M.

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**Definition 9**

Let *M*∈[ *X*]^{m}. The power mset *P*(*M*) of *M* is the set of all submsets of *M*. We have *N*∈*P*(*M*) if and only if *N*⊆*M*. If *N*=*ϕ*, then *N*∈^{1}*P*(*M*), and if *N*≠*ϕ*, then *N*∈^{k}*P*(*M*) such that \(k=\prod _{z}\dbinom {|[M]_{z}|}{|[N]_{z}|}\), the product \(\prod _{z}\) is taken over distinct elements of the mset *N* and |[*M*]_{z}|=*m**iff* *z*∈^{m}*M*, |[*N*]_{z}|=*n* *i**f**f* *z*∈^{n}*N*, then\(\dbinom {|[M]_{z}|}{|[N]_{z}|}=\dbinom {m}{n}=\frac {m!}{n!(m-n)!}\).

The power set of an mset is the support set of the power mset and is denoted by *P*^{∗}(*M*). The following theorem shows the cardinality of the power set of an mset.

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**Definition 10**

Let *M*_{1} and *M*_{2} be two msets drawn from a set *X*, then the Cartesian product of *M*_{1} and *M*_{2} is defined as *M*_{1}×*M*_{2}={(*m*/*x*,*n*/*y*)/*m**n*:*x*∈^{m}*M*_{1}, *y*∈^{n}*M*_{2}}.

Here, the entry (*m*/*x*,*n*/*y*)/*m**n* in *M*_{1}×*M*_{2} denotes *x* is repeated *m* times in *M*_{1}, *y* is repeated *n* times in *M*_{2}, and the pair (*x*,*y*) is repeated *mn* times in *M*_{1}×*M*_{2}.

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**Definition 11**

A submset *R* of *M*_{1}×*M*_{2} is said to be an mset relation on *M* if every member (*m*/*x*,*n*/*y*) of *R* has a count, the product of *C*_{1}(*x*,*y*) and *C*_{2}(*x*,*y*). *m*/*x* related to *n*/*y* is denoted by (*m*/*x*)*R*(*n*/*y*). The domain of the mset relation *R* on *M* is defined as follows:

$$Dom(R)=\{x\in^{r}M: \exists\;y\in^{s}M\;such\;that\;(r/x)R(s/y)\}, \;where$$

$$C_{Dom\;(R)}(x)=Sup\{C_{1}(x,y):x\in^{r}M\}.$$

Also, the range of the mset relation *R* on *M* is defined as follows:

$$Ran(R)=\{y\in^{s}M: \exists\;x\in^{r}M\;such\;that\;(r/x)R(s/y)\},\; where$$

$$C_{Ran\;(R)}(y)=Sup\{C_{2}(x,y):y\in^{s}M\}.$$

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**Definition 12**

An mset relation *f* is called an mset function if for every element *m*/*x* in Dom *f*, there is exactly one *n*/*y* in Ran *f* such that (*m*/*x*,*n*/*y*) is in *f* with the pair occurring as the product of *C*_{1}(*x*,*y*) and *C*_{2}(*x*,*y*).

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**Definition 13**

An mset function *f* is one-one (injective) if no two elements in Dom *f* have the same image under *f* with *C*_{1}(*x*,*y*)≤*C*_{2}(*x*,*y*) for all (*x*,*y*) in *f*, i.e., if *m*_{1}/*x*_{1},*m*_{2}/*x*_{2} in Dom *f* and *m*_{1}/*x*_{1}≠*m*_{2}/*x*_{2} implies that *f*(*m*_{1}/*x*_{1})≠*f*(*m*_{2}/*x*_{2}). Thus, the one-one mset function is the mapping of the distinct elements of the domain to the distinct elements of the range.

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**Definition 14**

An mset function *f* is onto (surjective) if Ran *f* is equal to co-dom *f* and *C*_{1}(*x*,*y*)≥*C*_{2}(*x*,*y*) for all (*x*,*y*) in *f*. It may be noted that images of distinct elements of the domain need not be distinct elements of the range.

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**Definition 15**

Let *M* be an mset drawn from a set *X* and *τ*⊆*P*^{∗}(*M*). Then, *τ* is called an mset topology if *τ* satisfies the following properties:

- (1)
*ϕ* and M are in *τ*.

- (2)
The union of the elements of any subcollection of *τ* is in *τ*.

- (3)
The intersection of the elements of any finite subcollection of *τ* is in *τ*.

An mset topological space is an ordered pair (*M*,*τ*) consisting of an mset *M* and an mset topology *τ*⊆*P*^{∗}(*M*) on *M*. Note that *τ* is an ordinary set whose elements are msets and the mset topology is abbreviated as an *M*-topology. Also, a submset *U* of *M* is an open mset of *M* if *U* belongs to the collection *τ*. Moreover, a submset *N* of *M* is closed mset *M*⊖*N* is open mset.

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**Definition 16**

Let (*M*,*τ*) be an *M*-topological space and *N* be a submset of *M*. Then, the interior of *N* is defined as the mset union of all open msets contained in *N* and is denoted by *N*^{o}; that is, *N*^{o}=∪{*V*⊆*M*:*V* is an open mset and *V*⊆*N*} and \(C_{N^{o}}(x)=\max \{C_{V}(x): V\subseteq N\}\).

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**Definition 17**

Let (*M*,*τ*) be an *M*-topological space and *N* be a submset of *M*. Then, the closure of *N* is defined as the mset intersection of all closed msets containing *N* and is denoted by \(\overline {N}\); that is, \(\overline {N}=\cap \{K\subseteq M: K\) is a closed mset and *N*⊆*K*} and \(C_{\overline {N}}(x)=\min \{C_{K}(x): N\subseteq K\}\).

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**Definition 18**

An mset *M* is called simple if all its elements are the same. For example, {*k*/*x*}. In addition, *k*/*x* is called simple multipoint (for short mpoint).

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**Definition 19**

Let (*M*,*τ*) be a *M*-topological space, *x*∈^{k}*M*, and *N*⊆*M*. Then, *N* is said to be a neighborhood of *k*/*x* if there is an open mset *V* in *τ* such that *x*∈^{k}*V* and *C*_{V}(*y*)≤*C*_{N}(*y*) for all *y*≠*x* that is, \(\mathcal {N}_{k/x}=\{N\subseteq M: \exists \;V\in \tau \;\)such that *x*∈^{k}*V* *and* *C*_{V}(*y*)≤*C*_{N}(*y*) for all *y*≠*x*} is the collection of all *τ*-neighborhood of *k*/*x*.