 Original Research
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Multiset filters
Journal of the Egyptian Mathematical Society volume 27, Article number: 51 (2019)
Abstract
A multiset is a collection of objects in which repetition of elements is essential. This paper is an attempt to generalize the notion of filters in the multiset context. In addition, many deviations between multiset filters and ordinary filters have been presented. The relation between multiset filter and multiset ideal has been mentioned. Many properties of multiset filters, multiset ultrafilters, and convergence of multiset filters have been introduced. Also, the notions of basis and subbasis have been mentioned in the multiset context. Finally, several examples have been studied.
Introduction
In classical set theory, a set is a welldefined collection of distinct objects. If repeated occurrences of any object is allowed in a set, then a mathematical structure is known as multiset (mset [1] or bag [2], for short). Thus, an mset differs from a set in the sense that each element has a multiplicity and a natural number not necessarily one that indicates how many times it is a member of the mset. One of the most natural and simplest examples is the mset of prime factors of a positive integer n. The number 400 has the factorization 400=2^{4}5^{2} which gives the mset {2,2,2,2,5,5}. Also, the cubic equation x^{3}−5x^{2}+3x+9=0 has roots 3,3, and − 1 which give the mset {3,3,− 1}.
Classical set theory is a basic concept to represent various situations in mathematical notation where repeated occurrences of elements are not allowed. But in various circumstances, repetition of elements becomes mandatory to the system. For example, in a graph with loops, there are many hydrogen atoms, many water molecules, many strands of identical DNA, etc. This leads to effectively three possible relations between any two physical objects: they are different, they are the same but separate, or they are coinciding and identical. For example, ammonia NH_{3} has with three hydrogen atoms, say H, H, and H, and one nitrogen atom, say N. Clearly, H and N are different. However H, H, and H are the same but separate, while H and H are coinciding and identical. There are many other examples, for instance, carbon dioxide CO_{2}, sulfuric acid H_{2}SO_{4}, and water H_{2}O.
This paper is an attempt to explore the theoretical aspects of msets by extending the notions of filters, ultrafilters, and convergence of filters to the mset context. The “Preliminaries and basic definitions” section has a collection of all basic definitions and notions for further study. In the “On multiset topologies” section, examples of new mset topologies are introduced. In the “Filters in multiset context” section, the notion of mset filters has been introduced. Further, many properties of this notion have been mentioned. In the “Basis and subbasis in multiset filters” section, basis and subbasis of mset filters are mentioned. In the “Multiset ultrafilter” section, the concept of mset ultrafilter has been presented and several examples and properties of this notion are introduced. In the “Convergence of multiset filters” section, convergence of mset filters and its properties are studied.
Preliminaries and basic definitions
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 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.
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).
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.
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^{∗}.
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^{∗}.
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 nonzero multiplicity as in M, i.e., M^{∗}=N^{∗} with C_{N}(x)≤C_{M}(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∈^{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}=miff z∈^{m}M, [N]_{z}=n iff z∈^{n}N, then\(\dbinom {[M]_{z}}{[N]_{z}}=\dbinom {m}{n}=\frac {m!}{n!(mn)!}\).
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 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)/mn:x∈^{m}M_{1}, y∈^{n}M_{2}}.
Here, the entry (m/x,n/y)/mn 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}.
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:
Also, the range of the mset relation R on M is defined as follows:
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).
Definition 13
An mset function f is oneone (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 oneone 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 codom 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.
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 Mtopology. 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 Mtopological 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 \(C_{N^{o}}(x)=\max \{C_{V}(x): V\subseteq N\}\).
Definition 17
Let (M,τ) be an Mtopological 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 Mtopological 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 τneighborhood of k/x.
On multiset topologies
Theorem 1
Let f:M_{1}→M_{2} be an mset function, V⊆M_{2} and N⊆M_{1}. Then:
 (1)
f^{−1}(M_{2}⊖V)=f^{−1}(M_{2})⊖f^{−1}(V).
 (2)
N⊆f^{−1}(f(N)), equality holds if f is oneone.
 (3)
f(f^{−1}(V))⊆V, equality holds if f is onto.
Proof
 (1)
Let x∈^{k}f^{−1}(M_{2}⊖V). Hence f(k/x)∈(M_{2}⊖V). So f(k/x)∉V; that is, k/x∈f^{−1}(M_{2}) and k/x∉f^{−1}(V). Thus, f^{−1}(M_{2}⊖V)⊆f^{−1}(M_{2})⊖f^{−1}(V). Also, let x∈^{k}f^{−1}(M_{2})⊖f^{−1}(V). It follows that f(k/x)∈M_{2} and f(k/x)∉V. Consequently, x∈^{k}f^{−1}(M_{2}⊖V). Therefore, f^{−1}(M_{2})⊖f^{−1}(V)⊆f^{−1}(M_{2}⊖V). Hence, f^{−1}(M_{2}⊖V)=f^{−1}(M_{2})⊖f^{−1}(V).
 (2)
Let x∈^{k}N. Hence, f(k/x)∈f(N). So x∈^{k}f^{−1}(f(N)), and hence, N⊆f^{−1}(f(N)). Now, let f be oneone and x∈^{k}f^{−1}(f(N)). It follows that f(k/x)∈f(N). So there exists y∈^{r}N such that f(k/x)=f(r/y). Since f is oneone, then k/x=r/y. Therefore, x∈^{k}N. Thus, if f is oneone, then f^{−1}(f(N))⊆N.
 (3)
Let x∈^{k}f(f^{−1}(V)). It follows that there exists y∈^{r}f^{−1}(V) such that f(k/x)=f(r/y) and f(r/y)∈V. So f(k/x)∈V. Thus, f(f^{−1}(V))⊆V. Also, if f is onto and x∈^{k}V. Hence, f^{−1}(k/x)∈f^{−1}(V), f is onto, so k/x∈f(f^{−1}(V)). Therefore, if f is onto, then V⊆f(f^{−1}(V)).
□
Theorem 2
Let N_{1} and N_{2} be submsets of an mset M. Then:
 (1)
If \(C_{(N_{1}\cap N_{2})}(x)=0\) for all x∈M^{∗}, then \(C_{N_{1}}(x)\leq C_{(M\ominus N_{2})}(x)\) for all x∈M^{∗}.
 (2)
\(C_{N_{1}}(x)\leq C_{N_{2}}(x)\Leftrightarrow C_{(M\ominus N_{2})}(x)\leq C_{(M\ominus N_{1})}(x)\) for all x∈M^{∗}.
Proof
 (1)
If \(\phantom {\dot {i}\!}C_{(N_{1}\cap N_{2})}(x)=0\) for all x∈M^{∗}. Since \(\phantom {\dot {i}\!}C_{(N_{1}\cap N_{2})}(x)=\min \{C_{N_{1}}(x), C_{N_{2}}(x)\}\), then \(C_{N_{1}}(x)=0\) or \(C_{N_{2}}(x)=0\) for all x∈M^{∗}. It follows that \(\phantom {\dot {i}\!}C_{N_{1}}(x)+C_{N_{2}}(x)\leq C_{M}(x)\) for all x∈M^{∗}, and hence, \(\phantom {\dot {i}\!}C_{N_{1}}(x)\leq C_{M}(x)C_{N_{2}}(x)=C_{(M\ominus N_{2})}(x)\) for all x∈M^{∗}, then the result.
 (2)
\(C_{N_{1}}(x)\leq C_{N_{2}}(x)\Leftrightarrow C_{N_{2}}(x)\leq C_{N_{1}}(x)\Leftrightarrow C_{M}(x)C_{N_{2}}(x)\leq C_{M}(x)C_{N_{1}}(x)\Leftrightarrow C_{(M\ominus N_{2})}(x)\leq C_{(M\ominus N_{1})}(x)\) for all x∈M^{∗}.
□
The following example shows that the converse of Theorem 2 is not true in general.
Example 1
Let M={2/a,4/b,5/c}, N_{1}={1/a,1/b,2/c}, and N_{2}={1/a,1/b}. Hence, M⊖N_{2}={1/a,3/b,5/c}. It is clear that N_{1}⊆M⊖N_{2} but N_{1}∩N_{2}={1/a,1/b}.
The following example shows that N_{1}⊖N_{2}≠N_{1}∩(M⊖N_{2}) in general.
Example 2
Let M={3/x,4/y}, N_{1}={2/x,3/y}, and N_{2}={1/x,2/y}. Hence, M⊖N_{2}={2/x,2/y}, N_{1}⊖N_{2}={1/x,1/y}, and N_{1}∩(M⊖N_{2})={2/x,2/y}.
Definition 20
Let X be an infinite set. Then, M={k_{α}/x_{α}:α∈Λ} be an infinite mset drawn from X. That is, the infinite mset M drawn from X is denoted by M={k_{1}/x_{1},k_{2}/x_{2},k_{3}/x_{3},… }.
Notation 1
The mset space \([\!X]_{\infty }^{m}\) is the set of all infinite msets whose elements are in X such that no element in the mset occurs more than m times.
It may be noted that the following examples of mset topologies are not tackled before.
Example 3
Let \(M\in [\!X]_{\infty }^{m}\) and {k_{0}/x_{0}} be a simple submset of M. Then, the collection \(\tau _{(k_{0}/x_{0})}=\{V\subseteq M: C_{V}(x_{0})\geq k_{0}\}\cup \{\emptyset \}\) is an Mtopology on M called the particular point Mtopology.
Example 4
Let \(M\in [\!X]_{\infty }^{m}\) and {k_{0}/x_{0}} be a simple submset of M. Then, the collection \(\tau _{k_{0}/x_{0}}=\{V\subseteq M: C_{V}(x_{0})< k_{0}\}\cup \{M\}\) is an Mtopology on M called the excluded point Mtopology.
Example 5
Let \(M\in [\!X]_{\infty }^{m}\). Then, the collection τ={V⊆M:M⊖V is finite }∪{∅} is an Mtopology on M called the cofinite Mtopology.
Example 6
Let \(M\in [\!X]_{\infty }^{m}\) and N be a submset of M. Then, the collection τ_{(N)}={V⊆M:C_{N}(x)≤C_{V}(x) for all x∈M^{∗}}∪{∅} is an Mtopology on M.
Example 7
Let \(M\in [\!X]_{\infty }^{m}\) and N be a submset of M. Then, the collection τ_{N}={V⊆M:C_{N}(x)≥C_{V}(x) for all x∈M^{∗}}∪{M} is an Mtopology on M.
Filters in multiset context
Definition 21
An mset filter \(\mathcal {F}\) on an mset M is a nonempty collection of nonempty submsets of M with the properties: \(({\mathcal {M}\mathcal {F}}_{1})\)\(\phi \not \in \mathcal {F}\), \(({\mathcal {M}\mathcal {F}}_{2})\) If \(N_{1}, N_{2}\in \mathcal {F}\), then \(N_{1}\cap N_{2}\in \mathcal {F}\), \(({\mathcal {M}\mathcal {F}}_{3})\) If \(N_{1}\in \mathcal {F}\) and \(C_{N_{1}}(x)\leq C_{N_{2}}(x)\) for all x∈M^{∗}, then \(N_{2}\in \mathcal {F}\).
It should be noted that \(\mathcal {F}\) is an ordinary set whose elements are msets and the multiset filter is abbreviated as an Mfilter.
Proposition 1
Let \(\mathcal {F}\) be an Mfilter on a nonempty mset M. Then:
 (1)
\(M\in \mathcal {F}\),
 (2)
Finite intersections of members of \(\mathcal {F}\) are in \(\mathcal {F}\).
Proof
The result follows immediately from Definition 21. □
Remark 1
It should be noted that the collection of complements of msets in a proper Mfilter is a nonempty collection closed under the operations of subsets and finite unions. Such a collection is called Mideal [17].
Example 8
PF(M) is an Mfilter on M.
Example 9
P^{∗}(M) is not an Mfilter. For one thing, the empty set belongs to it. Secondly, it contains the disjoint msets.
Example 10
Let \(\mathcal {F}=\{M\}\). Then, \(\mathcal {F}\) is an Mfilter. This is the smallest Mfilter one can define on M and is called the indiscrete Mfilter on M.
Example 11
Let x∈^{k}M and <k/x>={N⊆M:k/x∈N}. Then, <k/x> is an Mfilter called the principle Mfilter at <k/x>.
Example 12
More generally, let N be a nonempty submset of M and <N>={G⊆M:N⊆G}. Then, <N> is an Mfilter called the principle Mfilter at N. In addition to that, the indiscrete Mfilter is the principle Mfilter at M.
Example 13
Let M be an infinite mset and \(\mathcal {F}=\{N\subseteq M: N^{c} \)is a finite }. Then, \(\mathcal {F}\) is called the cofinite Mfilter on M.
Example 14
Let (M,τ) be an Mtopological space and x∈^{k}M. Then, \(\mathcal {N}_{k/x}\) is an Mfilter on M.
It should be noted that the Mfilter may contain the submset and it is complement because the intersection between submset and its complement is not necessary empty.
Example 15
Let M={2/a,3/b} and \(\mathcal {F}=\{M, \{1/a\}, \{2/a\}, \{1/a, 1/b\}, \{1/a, 2/b\}, \{1/a, 3/b\}, \{2/a, 1/b\}, \{2/a, 2/b\}\}\). It is clear that \(\mathcal {F}\) is an Mfilter and {1/a} and its complement {1/a,3/b} belong to \(\mathcal {F}\).
Definition 22
Let M be a nonempty mset and \(\mathcal {F}_{1}\), \(\mathcal {F}_{2}\) be two Mfilters on M. Then, \(\mathcal {F}_{1}\) is said to be coarser or smaller than \(\mathcal {F}_{2}\), denoted by \(\mathcal {F}_{1}\leq \mathcal {F}_{2}\), if \(\mathcal {F}_{1}\subseteq \mathcal {F}_{2}\), or alternatively \(\mathcal {F}_{2}\) is said to be finer or stronger than \(\mathcal {F}_{1}\).
Theorem 3
Let M be an mset and \(\{\mathcal {F}_{i}\}\), i∈I be a nonempty family of Mfilters on M. Then, \(\mathcal {F}=\cap _{i\in I}\mathcal {F}_{i}\) is an Mfilter on M.
Proof
Since \(M\in \mathcal {F}_{i}\) for each i∈I, hence \(M\in \cap _{i\in I}\mathcal {F}_{i}\); that is, \(M\in \mathcal {F}\). Moreover, \(({\mathcal {M}\mathcal {F}}_{1})\) implies \(\phi \not \in \mathcal {F}_{i}\) for each i∈I. Therefore, \(\mathcal {F}\) be a nonempty collection of a nonempty submsets of M. Let \(N_{1}, N_{2}\in \mathcal {F}\), then \(N_{1}, N_{2}\in \mathcal {F}_{i}\) for each i∈I. Since \(\mathcal {F}_{i}\) is an Mfilter for each i∈I, hence \(({\mathcal {M}\mathcal {F}}_{2})\) implies \(N_{1}\cap N_{2}\in \mathcal {F}_{i}\) for each i∈I. Thus, \(N_{1}\cap N_{2}\in \mathcal {F}\). Now let \(N_{1}\in \mathcal {F}\) and \(C_{N_{1}}(x)\leq C_{N_{2}}(x)\) for all x∈M^{∗}. It follows that \(N_{1}\in \mathcal {F}_{i}\) for each i∈I. Hence,\(({\mathcal {M}\mathcal {F}}_{2})\) implies that \(N_{2}\in \mathcal {F}_{i}\) for each i∈I. Therefore, \(N_{2}\in \mathcal {F}\), and hence, the result follows. □
The following example shows that the union of two Mfilters on a nonempty mset M is not necessarily an Mfilter on M.
Example 16
Let \(M=\{3/a, 4/b, 2/c, 5/d\},\;\mathcal {F}_{1}=\{M, \{3/a, 4/b, 2/c\}\}\), and \(\mathcal {F}_{2}=\{M, \{3/a, 4/b, 5/d\}\}\). Then, \(\mathcal {F}_{1}\cup \mathcal {F}_{2}=\{M, \{3/a, 4/b, 2/c\}, \{3/a, 4/b, 5/d\}\}\). Although \(\mathcal {F}_{1}\) and \(\mathcal {F}_{2}\) are two Mfilters on M, \(\mathcal {F}_{1}\cup \mathcal {F}_{2}\) is not Mfilter. Since \(\{3/a, 4/b, 2/c\}, \{3/a, 4/b, 5/d\}\in \mathcal {F}_{1}\cup \mathcal {F}_{2}\), but \(\{3/a, 4/b, 2/c\}\cap \{3/a, 4/b, 5/d\}=\{3/a, 4/b\}\not \in \mathcal {F}_{1}\cup \mathcal {F}_{2}\).
Basis and subbasis in multiset filters
Definition 23
Let \(\mathcal {B}\) be a nonempty collection of a nonempty submsets of M. Then, \(\mathcal {B}\) is called an Mfilter basis on M if \(({\mathcal {M}\mathcal {B}}_{1})\)\(\phi \not \in \mathcal {B}\), \(({\mathcal {M}\mathcal {B}}_{2})\) If \(B_{1}, B_{2}\in \mathcal {B}\), then there exists a \(B\in \mathcal {B}\) such that \(C_{B}(x)\leq C_{(B_{1}\cap B_{2})}(x)\) for all x∈M^{∗}.
Theorem 4
Let \(\mathcal {B}\) be an Mfilter basis on M, and \(\mathcal {F}\) consists of all msets which are super msets in \(\mathcal {B}\); that is, \(\mathcal {F}=\{N\subseteq M: \;\forall \;x\in M^{*}\;C_{N}(x)\geq C_{N_{1}}(x),\;\phantom {\dot {i}\!}\)for some\(\;N_{1}\in \mathcal {B}\}. \)Then, \(\mathcal {F}\) is an Mfilter on M. Furthermore, it is the smallest Mfilter which contains \(\mathcal {B}\). It is called the Mfilter generated by \(\mathcal {B}\).
Proof
Since \(\mathcal {F}\) consists of all msets which are super msets in \(\mathcal {B}\), hence every member of \(\mathcal {B}\) is also a member of \(\mathcal {F}\). Consequently, \(\mathcal {B}\subseteq \mathcal {F}\) and hence \(\mathcal {F}\neq \phi \). Since \(\mathcal {F}\) contains all submsets of M which contain a member of \(\mathcal {B}\) and \(\phi \not \in \mathcal {B}\), hence \(\phi \not \in \mathcal {F}\). Thus, \(\mathcal {F}\) satisfies \(({\mathcal {M}\mathcal {F}}_{1})\). To prove that \(\mathcal {F}\) satisfies \(({\mathcal {M}\mathcal {F}}_{2})\), let \(N_{1}, N_{2}\in \mathcal {F}\). Hence, for all x∈M^{∗}, \(C_{N_{1}}(x)\geq B_{1}(x)\) and \(C_{N_{2}}(x)\geq C_{B_{2}}(x)\phantom {\dot {i}\!}\) for some \(B_{1}, B_{2}\in \mathcal {B}\). It follows that there exists \(B\in \mathcal {B}\) such that \(C_{B}(x)\leq C_{(B_{1}\cap B_{2})}(x)\) for all x∈M^{∗} and hence \(C_{(N_{1}\cap N_{2})}(x)\geq C_{(B_{1}\cap B_{2})}(x)\geq C_{B}(x)\) for all x∈M^{∗}. Consequently, \(N_{1}\cap N_{2}\in \mathcal {F}\). For \(({\mathcal {M}\mathcal {F}}_{3})\), let \(N_{1}\in \mathcal {F}\) and \(C_{N_{1}}(x)\leq C_{N_{2}}(x)\) for all x∈M^{∗}. It follows that for all \(x\in M^{*} C_{N_{1}}(x)\geq C_{B}(x)\phantom {\dot {i}\!}\) for some \(B\in \mathcal {B}\). Therefore, for all \(x\in M^{*} C_{N_{2}}(x)\geq C_{N_{1}}(x)\geq C_{B}(x)\phantom {\dot {i}\!}\) for some \(B\in \mathcal {B}\). Thus, \(N_{2}\in \mathcal {F}\). Hence, \(\mathcal {F}\) is an Mfilter on M. Now, let \(\mathcal {F}_{1}\) be an Mfilter which contains \(\mathcal {B}\) such that \(\mathcal {F}_{1}\leq \mathcal {F}\). Let \(N\in \mathcal {F}\). It follows that for all \(x\in M^{*}\;C_{N}(x)\geq C_{N_{1}}(x)\phantom {\dot {i}\!}\), for some \(N_{1}\in \mathcal {B}\). This result, combined with \(N_{1}\in \mathcal {F}_{1}\) and \(({\mathcal {M}\mathcal {F}}_{3})\), implies \(N\in \mathcal {F}_{1}\). Hence, \(\mathcal {F}\leq \mathcal {F}_{1}\). Therefore, \(\mathcal {F}=\mathcal {F}_{1}\). Thus, \(\mathcal {F}\) is the smallest Mfilter which contains \(\mathcal {B}\). □
Example 17
Every Mfilter is trivially an Mfilter basis of itself.
Example 18
\(\mathcal {B}=\{k/x\}\) is an Mfilter basis and generates the principle Mfilter at k/x.
Example 19
\(\mathcal {B}=\{N\}\) is an Mfilter basis and generates the principle Mfilter at N.
Example 20
Let M={k_{1}/x_{1},k_{2}/x_{2},k_{3}/x_{3},…,k_{m}/x_{n}}. Then, \(\mathcal {B}=\{1/x_{1}, 1/x_{2}, 1/x_{3},\dots, 1/x_{n}\}\) is an Mfilter basis and generates PF(M).
Theorem 5
Let M be a nonempty mset, \(\mathcal {B}\) an Mfilter basis which generates \(\mathcal {F}_{1}\), and \(\mathcal {B^{*}}\) an Mfilter basis which generates \(\mathcal {F}_{2}\). Then, \(\mathcal {F}_{1}\leq \mathcal {F}_{2}\) if and only if every member of \(\mathcal {B}\) contains a member of \(\mathcal {B^{*}}\).
Proof
Suppose \(\mathcal {F}_{1}\leq \mathcal {F}_{2}\phantom {\dot {i}\!}\) and \(B\in \mathcal {B}\). Since \(\mathcal {B}\) is an Mfilter basis which generates \(\mathcal {F}_{1}\), then \(B\in \mathcal {F}_{1}\). Since \(\mathcal {F}_{1}\leq \mathcal {F}_{2}\), thus \(B\in \mathcal {F}_{2}\), which implies that there exists \(B^{*}\in \mathcal {B^{*}}\phantom {\dot {i}\!}\) such that \(C_{B^{*}}(x)\leq C_{B}(x)\phantom {\dot {i}\!}\) for all x∈M^{∗}. Therefore, every member of \(\mathcal {B}\) contains a member of \(\mathcal {B^{*}}\). On the other hand, let every member of \(\mathcal {B}\) contain a member of \(\mathcal {B^{*}}\) and \(F\in \mathcal {F}_{1}\). Since \(\mathcal {B}\) is an Mfilter basis which generates \(\mathcal {F}_{1}\), it follows that there exists \(B\in \mathcal {B}\) such that C_{B}(x)≤C_{F}(x) for all x∈M^{∗}. From the assumption, there exists \(B^{*}\in \mathcal {B^{*}}\) such that \(C_{B^{*}}(x)\leq C_{B}(x)\leq C_{F}(x)\phantom {\dot {i}\!}\) for all x∈M^{∗}. This result, combined with \(\mathcal {B^{*}}\) is an Mfilter basis which generates \(\mathcal {F}_{2}\), implies \(F\in \mathcal {F}_{2}\). Consequently, \(\mathcal {F}_{1}\leq \mathcal {F}_{2}\). □
Definition 24
The two Mfilter basis (Mfilter subbasis) are \(\mathcal {F}\)equivalent if they generate the same Mfilter.
Theorem 6
Let \(\mathcal {B}\) and \(\mathcal {B^{*}}\) be an Mfilter basis on a nonempty mset M. Then, \(\mathcal {B}\) and \(\mathcal {B^{*}}\) are equivalent if and only if every member of \(\mathcal {B}\) contains a member of \(\mathcal {B^{*}}\) and every member of \(\mathcal {B^{*}}\) contains a member of \(\mathcal {B}\).
Proof
The result follows immediately from Theorem 4. □
Theorem 7
Let M_{1} and M_{2} be two nonempty msets drawn from X and Y, respectively, f:M_{1}→M_{2} be an mset function, \(\mathcal {B}\) be an Mfilter basis on M_{1}, and \(\mathcal {B^{*}}\) be an Mfilter basis on M_{2}. Then:
 (1)
\(K_{1}=\{f(B) : B\in \mathcal {B}\}\) is an Mfilter basis on M_{2}.
 (2)
If every member of \(\mathcal {B^{*}}\) intersects f(M_{1}), then \(K_{2}=\{f^{1}(B^{*}) : B^{*}\in \mathcal {B^{*}}\}\) is an Mfilter basis on M_{1}.
Proof
 (1)
Since \(\mathcal {B}\) is an Mfilter basis on M_{1}. It follows that \(\mathcal {B}\neq \phi \). So, K_{1}≠ϕ. For \(({\mathcal {M}\mathcal {B}}_{1})\), since \(\phi \not \in \mathcal {B}\), hence ϕ∉K_{1}. To prove \(({\mathcal {M}\mathcal {B}}_{2})\), let \(f(B_{1}), f(B_{2})\in \mathcal {B}\) such that \(B_{1}, B_{2}\in \mathcal {B}\). Since \(\mathcal {B}\) is an Mfilter basis on M_{1}, it follows that there exists \(B\in \mathcal {B}\) such that \(C_{B}(x)\leq C_{(B_{1}\cap B_{2})}(x)\) for all x∈M^{∗}. Thus, \(C_{f(B)}(y)\leq C_{f(B_{1}\cap B_{2})}(y)\leq C_{f(B_{1})\cap f(B_{2})}(y)\) for all y∈Y. Therefore, there exists f(B)∈K_{1} such that \(C_{f(B)}(y)\leq C_{f(B_{1})\cap f(B_{2})}(y)\) for all y∈Y. Hence, K_{1} is an Mfilter basis on M_{2}.
 (2)
The proof is similar to part (1).
□
Multiset ultrafilter
Definition 25
An Mfilter \(\mathcal {F}\) is called an mset ultrafilter on M, Multrafilter, if there is no strictly finer Mfilter than \(\mathcal {F}\). That is, if \(\mathcal {F}^{*}\) is an Multrafilter and \(\mathcal {F^{*}}\geq \mathcal {F}\), then \(\mathcal {F^{*}}=\mathcal {F}\).
Example 21
Let M={2/a,3/b}. Then, \(\mathcal {F}_{1}=\{M, \{1/a\},\{2/a\},\{1/a, 1/b\}, \{1/a, 2/b\},\{1/a, 3/b\}, \{2/a, 1/b\},\{2/a, 2/b\}\}\) and \(\mathcal {F}_{2}=\{M, \{1/b\}, \{2/b\}, \{3/b\}, \{1/a, 1/b\}, \{2/a, 1/b\}, \{1/a, 2/b\}, \{2/a, 2/b\}, \{1/a, 3/b\}, \{2/a, 3/b\}\}\) are Multrafilters on M.
Theorem 8
Let M be a nonempty mset. An Mfilter \(\mathcal {F}\) is an Multrafilter if it contains all submsets of M which intersects every member of \(\mathcal {F}\).
Proof
Let \(\mathcal {F}\) be an Multrafilter on M and N be a submset of M such that F∩N≠ϕ for all \(F\in \mathcal {F}\). Now, we want to show that the collection \(\mathcal {F}^{*}=\{F^{*}: \forall \;x\in M^{*}\;C_{F^{*}}(x)\geq C_{(N\cap F)}(x)\;\)for some \(F\in \mathcal {F}\}\) is an Mfilter on M. For \(({\mathcal {M}\mathcal {F}}_{1})\), since C_{(N∩F)}(x)≥C_{ϕ}(x) for all x∈M^{∗}. Thus, \(\phi \not \in \mathcal {F}^{*}\). For \(({\mathcal {M}\mathcal {F}}_{2})\), let \(F_{1}^{*}\), \(F_{2}^{*}\in \mathcal {F}^{*}\phantom {\dot {i}\!}\). Hence, for all \(x\in M^{*} C_{F_{1}^{*}}(x)\geq C_{(N\cap F_{1})}(x)\) and \(C_{F_{2}^{*}}(x)\geq C_{(N\cap F_{2})}(x)\phantom {\dot {i}\!}\) for some \(F_{1},F_{2}\in \mathcal {F}\). It follows that for all \(x\in M^{*} C_{(F_{1}^{*}\cap F_{2}^{*})}(x)\geq C_{[N\cap (F_{1}\cap F_{2})]}(x)\). Therefore, \(F_{1}^{*}\cap F_{2}^{*}\in \mathcal {F}^{*}\). To prove that \(\mathcal {F}^{*}\) satisfies \(({\mathcal {M}\mathcal {F}}_{3})\), let \(F_{1}^{*}\in \mathcal {F}^{*}\) and \(\phantom {\dot {i}\!}C_{F_{1}^{*}}(x)\leq C_{F_{2}^{*}}(x)\) for all x∈M^{∗}. It follows that for all \(\phantom {\dot {i}\!}x\in M^{*} C_{F_{2}^{*}}(x)\geq C_{F_{1}^{*}}(x)\geq C_{(N\cap F)}(x)\) for some \(F\in \mathcal {F}\). Consequently, \(F_{2}^{*}\in \mathcal {F}^{*}\). Hence, \(\mathcal {F}^{*}\) is an Mfilter on M. Since C_{F}(x)≥C_{(N∩F)}(x) for all x∈M^{∗}, then \(\mathcal {F}^{*}\geq \mathcal {F}\). Since \(\mathcal {F}\) is an Multrafilter on M, it follows that \(\mathcal {F}^{*}=\mathcal {F}\). Moreover, \(N\in \mathcal {F}\) as for all x∈M^{∗}C_{N}(x)≥C_{(N∩F)}(x). □
Theorem 9
Let \(\mathcal {F}\) be an Multrafilter on a nonempty mset M. Then, for each N⊆M, either N or \(N^{c}\in \mathcal {F}\).
Proof
Let \(\mathcal {F}\) be an Multrafilter on M and N⊆M. If there exists \(F\in \mathcal {F}\) such that C_{(F∩N)}(x)=0 for all x∈M^{∗}, then Theorem 2 part (1) implies \(\phantom {\dot {i}\!}C_{F}(x)\leq C_{N^{c}}(x)\) for all x∈M^{∗}. Thus, \(N^{c}\in \mathcal {F}\). Otherwise, C_{(F∩N)}(x)>0 for all x∈M^{∗}. Thus, Theorem 8 implies \(N\in \mathcal {F}\), then the result. □
The following example shows that the converse of Theorem 9 is incorrect in general.
Example 22
Let M={2/a,3/b}. Then, \(\mathcal {F}=\{M, \{2/b\}, \{3/b\}, \{1/a, 2/b\}, \{1/a, 3/b\}, \{2/a, 2/b\}, \{2/a, 1/b\},\{2/a, 2/b\}\}\) is an Mfilter on M. Although for each N⊆M, either N or \(N^{c}\in \mathcal {F}\), \(\mathcal {F}\) is not Multrafilter. As \(\mathcal {F}^{*}=\{M, \{1/b\},\{2/b\}, \{3/b\}, \{1/a, 1/b\}, \{1/a, 2/b\}, \{1/a, 3/b\}, \{2/a, 1/b\}, \{2/a, 2/b\},\{2/a, 3/b\}\}\) is finer than \(\mathcal {F}\).
Theorem 10
Let \(\mathcal {F}\) be an Multrafilter on a nonempty mset M. Then, for each two nonempty submsets N_{1},N_{2} of M such that \(N_{1}\cup N_{2}\in \mathcal {F}\), either \(N_{1}\in \mathcal {F}\) or \(N_{2}\in \mathcal {F}\).
Proof
Assume \(N_{1}\cup N_{2}\in \mathcal {F}\) and \(N_{1}\in \mathcal {F}\) and \(N_{2}\in \mathcal {F}\). Define \(\phantom {\dot {i}\!}\mathcal {F}^{*}=\{G\subseteq M: G\cup N_{2}\in \mathcal {F}\}\). Now, we want to prove that \(\mathcal {F}^{*}\) is an Mfilter on M. Since \(N_{1}\cup N_{2}\in \mathcal {F}\), then \(\phantom {\dot {i}\!}N_{1}\in \mathcal {F}^{*}\). Hence, \(\mathcal {F}^{*}\neq \phi \). For \(({\mathcal {M}\mathcal {F}}_{1})\), since \(\phi \cup N_{2}=N_{2}\not \in \mathcal {F}^{*}\), it follows that \(\phi \not \in \mathcal {F}^{*}\). To prove that \(\mathcal {F}^{*}\) satisfies \(({\mathcal {M}\mathcal {F}}_{2})\), let \(\phantom {\dot {i}\!}G_{1}, G_{2}\in \mathcal {F}^{*}\). Hence, \(G_{1}\cup N_{2}\in \mathcal {F}\) and \(G_{2}\cup N_{2}\in \mathcal {F}\). Thus, \(\phantom {\dot {i}\!}(G_{1}\cup N_{2})\cap (G_{2}\cup N_{2})\in \mathcal {F}\). Therefore, \((G_{1}\cap G_{2})\cup N_{2}\in \mathcal {F}\). Hence, \(G_{1}\cap G_{2}\in \mathcal {F}\). For \(({\mathcal {M}\mathcal {F}}_{3})\), let \(\phantom {\dot {i}\!}G_{1}\in \mathcal {F}^{*}\) and \(C_{G_{1}}(x)\leq C_{G_{2}}(x)\phantom {\dot {i}\!}\) for all x∈M^{∗}. Hence, \(G_{1}\cup N_{2}\in \mathcal {F}\) and \(C_{(G_{1}\cup N_{2})}(x)\leq C_{(G_{2}\cup N_{2})}(x)\) for all x∈M^{∗}. Thus, \(G_{2}\cup N_{2}\in \mathcal {F}\). Hence, \(G_{2}\in \mathcal {F}^{*}\phantom {\dot {i}\!}\). Consequently, \(\mathcal {F}^{*}\) is an Mfilter on M. Let \(F\in \mathcal {F}\). Since \(\phantom {\dot {i}\!}C_{F}(x)\leq C_{(F\cup N_{2})}(x)\) for all x∈M^{∗}, then \(({\mathcal {M}\mathcal {F}}_{3})\) implies \(F\cup N_{2}\in \mathcal {F}\). Therefore, \(F\in \mathcal {F}^{*}\); that is, \(\mathcal {F}\leq \mathcal {F}^{*}\). But \(\mathcal {F}\) is an Multrafilter. Thus, there is a contradiction. Therefore, \(N_{1}\in \mathcal {F}\) or \(N_{2}\in \mathcal {F}\). □
The following example shows that the converse of Theorem 10 is wrong in general.
Example 23
Let M={3/a,4/b} and \(\mathcal {F}=\{M, \{3/a\}, \{3/a, 1/b\}, \{3/a, 2/b\}, \{3/a, 3/b\}\}\) be an Mfilter on M. Although for all N_{1},N_{2}⊆M such that \(N_{1}\cup N_{2}\in \mathcal {F}\), either \(N_{1}\in \mathcal {F}\) or \(N_{2}\in \mathcal {F}\), \(\mathcal {F}\) is not Multrafilter. As \(\mathcal {F}^{*}=\{M, \{3/a\}, \{4/b\}, \{3/a, 1/b\}, \{3/a, 2/b\}, \{3/a, 3/b\}, \{1/a, 4/b\}, \{2/a, 4/b\}\}\) is finer than \(\mathcal {F}\).
Convergence of multiset filters
Definition 26
Let (M,τ) be an Mtopological space and \(\mathcal {F}\) be an Mfilter on M. \(\mathcal {F}\) is said to τconverge to k/x (written \(\mathcal {F}\overset {\tau }{\longrightarrow }k/x\)) if \(\mathcal {N}_{k/x}\subseteq \mathcal {F}\); that is, if \(\mathcal {F}\geq \mathcal {N}_{k/x}\).
Example 24
For each mpoint, k/x, \(\mathcal {N}_{k/x}\) converges to k/x.
Example 25
Let τ be the cofinite Mtopology on M and \(\mathcal {F}\) be the cofinite Mfilter. Then, \(\mathcal {F}\) converges to each mpoint.
Example 26
Let (M,τ) be the indiscrete M topology and \(\mathcal {F}\) any Mfilter on M; then, \(\mathcal {F}\) converges to each mpoint.
Theorem 11
Let (M,τ) be an Mtopological space and \(\mathcal {F}\) and \(\mathcal {F}^{*}\) be Mfilters on M such that \(\mathcal {F}^{*}\geq \mathcal {F}\). If \(\mathcal {F}\overset {\tau }{\longrightarrow }k/x\), then \(\mathcal {F}^{*}\overset {\tau }{\longrightarrow }k/x\).
Proof
Since \(\mathcal {F}\overset {\tau }{\longrightarrow }k/x\) and \(\mathcal {F}^{*}\geq \mathcal {F}\), it follows that \(\mathcal {F}^{*}\geq \mathcal {F}\geq \mathcal {N}_{k/x}\). Hence, Definition 26 implies that \(\mathcal {F}^{*}\overset {\tau }{\longrightarrow }k/x\). □
Theorem 12
Let (M,τ_{1}) and (M,τ_{2}) be two Mtopological spaces such that τ_{2}≤τ_{1} and \(\mathcal {F}\) be an Mfilter on M such that \(\mathcal {F}\overset {\tau _{1}}{\longrightarrow }k/x\). Then, \(\mathcal {F}\overset {\tau _{2}}{\longrightarrow }k/x\).
Proof
Since τ_{2}≤τ_{1} and \(\mathcal {F}\overset {\tau _{1}}{\longrightarrow }k/x\), it follows that \(\mathcal {N}^{\tau _{2}}_{k/x}\leq \mathcal {N}^{\tau _{1}}_{k/x}\) and \(\mathcal {N}^{\tau _{1}}_{k/x}\leq \mathcal {F}\). Thus, \(\mathcal {N}^{\tau _{2}}_{k/x}\leq \mathcal {N}^{\tau _{1}}_{k/x}\leq \mathcal {F}\). Hence, Definition 26 implies \(\mathcal {F}\overset {\tau _{2}}{\longrightarrow }k/x\), then the result. □
Theorem 13
Let (M,τ) be an Mtopological space, then the following assertions are equivalent:
 (1)
\(\mathcal {F}\overset {\tau }{\longrightarrow }k/x\),
 (2)
Every Multrafilter containing \(\mathcal {F}\) converges to k/x.
Proof
On the one hand, let \(\mathcal {F}^{*}\) be an Multrafilter containing \(\mathcal {F}\); that is, \(\mathcal {F}\leq \mathcal {F}^{*}\). This result, combined with assertion (1), implies \(\mathcal {N}_{k/x}\leq \mathcal {F}\leq \mathcal {F}^{*}\). Thus, Definition 26 implies \(\mathcal {F}^{*}\overset {\tau }{\longrightarrow }k/x\). Hence, (1) implies (2). On the other hand, (2) implies that \(\mathcal {N}_{k/x}\) is contained in every Multrafilter containing \(\mathcal {F}\). Hence, \(\mathcal {N}_{k/x}\) is contained in the intersection of all Multrafilter containing \(\mathcal {F}\). This result, combined with \(\mathcal {F}\) is the intersection of all Multrafilter containing \(\mathcal {F}\), implies \(\mathcal {N}_{k/x}\leq \mathcal {F}\). Then, \(\mathcal {F}\overset {\tau }{\longrightarrow }k/x\). Hence, (2) implies (1). □
Theorem 14
Let (M,τ) be an Mtopological space and N be a nonempty submset of M; then, the following assertions are equivalent:
 (1)
N∈τ,
 (2)
If \(\mathcal {F}\overset {\tau }{\longrightarrow }k/x\) such that x∈^{k}N, then \(N\in \mathcal {F}\).
Proof
The first direction is a direct consequence of Definition 26 and assertion (1). On the other hand, let x∈^{m}N. Then Example 24 shows that \(\mathcal {N}_{m/x}\overset {\tau }{\longrightarrow }m/x\). Thus, assertion (2) implies that \(N\in \mathcal {N}_{m/x}\); that is, N is a neighborhood of m/x. Hence, N is a neighborhood for every x∈^{k}N. Thus, N∈τ; that is, (2) implies (1). □
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Abbreviations
 mset :

Multiset
 Mfilter :

multiset filter
 Multrafilter :

multiset ultrafilter
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Zakaria, A., John, S.J. & Girish, K.P. Multiset filters. J Egypt Math Soc 27, 51 (2019). https://doi.org/10.1186/s4278701900563
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Keywords
 Multiset
 Multiset filter
 Multiset ultrafilter
 Multiset topology
Mathematics Subject Classification (2010)
 54A05; 03E70