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].
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 CM 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={x1,x2,…,xn} with x appearing n times in M. It is denoted by x∈nM. The mset M drawn from the set X is denoted by M={k1/x1,k2/x2,…,kn/xn} where M is an mset with x1 appearing k1 times, x2 appearing k2 times, and so on. In Definition 10, CM(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 CM(x)=0 or 1 ∀ x∈X.
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 CM(x)≤CN(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 CM(x)≤CN(x) ∀ x∈X and there exists at least one element x∈X such that CM(x)<CN(x).
- (4)
P=M∪N if CP(x)= max{CM(x),CN(x)} for all x∈X.
- (5)
P=M∩N if CP(x)= min{CM(x),CN(x)} for all x∈X.
- (6)
Addition of M and N results is a new mset P=M⊕N such that CP(x)= min{CM(x)+CN(x),m} for all x∈X.
- (7)
Subtraction of M and N results in a new mset P=M⊖N such that CP(x)= max{CM(x)−CN(x),0} for all x∈X, where ⊕ and ⊖ represent mset addition and mset subtraction, respectively.
- (8)
An mset M is empty if CM(x)=0 ∀ x∈X.
- (9)
The support set of M denoted by M∗ is a subset of X and M∗={x∈X:CM(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).
Definition 3
Let M∈[ X]m and N⊆M. Then, the complement Nc of N in [ X]m is an element of [ X]m such that Nc=M⊖N.
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, CN(x)=CM(x) for every x∈N∗.
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., CN(x)=CM(x) for some x∈N∗.
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 CN(x)≤CM(x) for every x∈N∗.
Definition 7
Let M∈[X]m. The power whole mset of M denoted by PW(M) is defined as the set of all whole submsets of M.
Definition 8
Let M∈[ X]m. The power full msets of M, PF(M), is defined as the set of all full submsets of M. The cardinality of PF(M) is the product of the counts of the elements in M.
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∈1P(M), and if N≠ϕ, then N∈kP(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|=miff z∈mM, |[N]z|=n iff z∈nN, 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.
Definition 10
Let M1 and M2 be two msets drawn from a set X, then the Cartesian product of M1 and M2 is defined as M1×M2={(m/x,n/y)/mn:x∈mM1, y∈nM2}.
Here, the entry (m/x,n/y)/mn in M1×M2 denotes x is repeated m times in M1, y is repeated n times in M2, and the pair (x,y) is repeated mn times in M1×M2.
Definition 11
A submset R of M1×M2 is said to be an mset relation on M if every member (m/x,n/y) of R has a count, the product of C1(x,y) and C2(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:
$$Ran(R)=\{y\in^{s}M: \exists\;x\in^{r}M\;such\;that\;(r/x)R(s/y)\},\; where$$
$$C_{Ran\;(R)}(y)=Sup\{C_{2}(x,y):y\in^{s}M\}.$$
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 C1(x,y) and C2(x,y).
Definition 13
An mset function f is one-one (injective) if no two elements in Dom f have the same image under f with C1(x,y)≤C2(x,y) for all (x,y) in f, i.e., if m1/x1,m2/x2 in Dom f and m1/x1≠m2/x2 implies that f(m1/x1)≠f(m2/x2). Thus, the one-one mset function is the mapping of the distinct elements of the domain to the distinct elements of the range.
Definition 14
An mset function f is onto (surjective) if Ran f is equal to co-dom f and C1(x,y)≥C2(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.
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.
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 No; that is, No=∪{V⊆M:V is an open mset and V⊆N} and \(C_{N^{o}}(x)=\max \{C_{V}(x): V\subseteq N\}\).
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 \(C_{\overline {N}}(x)=\min \{C_{K}(x): N\subseteq K\}\).
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).
Definition 19
Let (M,τ) be a M-topological space, x∈kM, 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∈kV and CV(y)≤CN(y) for all y≠x that is, \(\mathcal {N}_{k/x}=\{N\subseteq M: \exists \;V\in \tau \;\)such that x∈kV and CV(y)≤CN(y) for all y≠x} is the collection of all τ-neighborhood of k/x.