Infinite multisets: Basic properties and cardinality Milen

: This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i


Introduction
In mathematics, a multiset (shortened to mset) is a modification of the concept of a set.Unlike sets, in msets each element may occur more than once.The multiplicity of an element is defined as the number of times (number of copies) it occurs in the mset.For example, in the mset ⦃, , , ⦄, the element  has multiplicity 2, and the two elements ,  have multiplicity 1.Multiple occurrences of an element in an mset are treated without preference [7].The same elements in the mset (in this case the elements , ) are indistinguishable from one another.
This research work presents the topic of infinite msets, their basic properties and cardinality.A definition of msets is given in the first part of the work and the basic operations and relations with them are presented.By means of the introduced operations with msets, the possibility of getting msets with negative and even fraction multiplicity is illustrated.The present article proposes broadening the concept of cardinality so that this concept is applicable to finite and infinite msets.A new property of multisets, 'm-cardinality', is defined using multiset functions.Relations are introduced which allow comparing finite and infinite msets and defining their m-cardinality.Some of the basic properties of these relations are formulated and proved.In the second part of the work the m-cardinality of infinite msets is investigated, comparisons are made, and the m-cardinality of concrete msets is determined.The existence of still unresearched hierarchies of infinities (corresponding to infinite msets) with m-cardinality less than the cardinality of a countably infinite set (ℵ 0 ), has been established.Therefore, between finite numbers and ℵ 0 there is an unlimited number of different transfinite m-cardinals, corresponding to infinite msets.

Definitions and properties
All elements of the viewed msets are taken in a fixed set  which we will call a universe.An element of  that does not belong to a given mset will naturally have multiplicity of 0 in this mset.
An mset  can be formally defined as an ordered pair where  * is a given set.Here   :  * →  ≠0 is a function giving each element its multiplicity from  * to a class  ≠0 of non-zero rational numbers and non-zero cardinal numbers (  ≠ 0).
For each  ∈  * ,   () is the characteristic value of  in  and indicates the number of occurrences of the element  in .(Hereon, msets will be denoted with ⦃ ⦄ brackets, and traditional sets will be denoted with { } brackets.)In fact,  * is the set of all distinguishable (distinct) elements of an mset  and is called its support or root set: In the present work, the root set  * is a countable set (finite or infinite).The mset will also be written in the following way:  = ⦃⦄   = ⦃ 1 ,  2 ,  3 , . . .⦄   ( 1 ),  ( 2 ),  ( 3 ),... .
The root set will also be denoted in the following way: Let  appear  times in the mset .It is denoted by  ∈  .Let ⦃⦄  denote the mset that contains exactly  copies of  and nothing else.
∈ * Let us look at msets  = ⦃ * ,   ⦄ and  = ⦃ * ,   ⦄.We will define the following relations: is an msubset of , denoted by  ⊆ , if Two msets  and  are equal, denoted by  = , if The union  of two msets  and  is defined as: The intersection  between two msets  and  is defined as: The sum  of these two msets  and  is defined as: The difference  of these two msets  and  is defined as: Since the range of the characteristic function is the class of all of non-zero rational numbers and non-zero cardinal numbers, there is no limitation on negative values of the multiplicity in the last operationthe difference of the msets.It is interesting to note that the limitation on negative values of the multiplicity can be ignored without considerably affecting the other properties and operations.These generalizations of the msets with negative multiplicity are examined in detail in many research works [2,3,5].
In this way, we can examine msets with a negative multiplicity (sometimes called a signed mset or, hybrid/shadow mset [3,9]), a fraction multiplicity and even a multiplicity which is a random real number (for example, real-valued msets, fuzzy sets [2]).The negative multiplicity is, in fact, the removing or deleting of an element from the mset.

2
. Here the element  has a fraction multiplicity which is equal to . We will have: The empty mset Ø = ⦃ ⦄ is the unique mset with an empty support and has total multiplicity 0. The empty mset is an msubset of any possible mset: ∀ (Ø ⊆ ).
We will call msets  ̅ and  ̅ absolute complement msets.Theorem 2.1.It is easy to prove that De Morgan properties are valid: Let  denote the empty mset, which contains null multiplicities for all elements in the respective support [5].We will have: The absolute complement can be formally defined as follows: (In the definition of the absolute complement, the empty mset has been used instead of the universe mset [5].) The Cartesian product of two msets  and  is defined as [6]: ⨯  = ⦃, , (, )⦄   ,  , , where  =     .
The entry of the form ⦃, , (, )⦄   ,  , denotes that  is repeated   times,  is repeated   times and the pair (, ) is repeated  times.The counts of the members of the domain and codomain vary with respect to the counts of the  coordinate and  coordinate in ⦃, , (, )⦄   ,  , .We introduce the notation  1 (, ) and  2 (, ), where  1 (, ) denotes the count of the first coordinate in the ordered pair (, ) and  2 (, ) denotes the count of the second coordinate in the ordered pair (, ).The Cartesian product of three or more nonempty msets is defined by generalizing the definition of the Cartesian product of two msets.
An mset relation  is called an mset function if for every element ⦃⦄  in Dom , there is exactly one ⦃⦄  in Ran  such that ⦃, ⦄ , is in  with the pair occurring only  1 (, ) times, [6].For functions between arbitrary msets, it is essential that images of indistinguishable elements of the domain must be indistinguishable elements of the range but the images of the distinct elements of the domain need not be distinct elements of the range.
Most authors [1,4,6,8] identify the cardinality of the mset with its total multiplicity.But the two notions have different significance and meaning.In the set theory, if two sets have the same cardinalities, then a bijection (one-to-one correspondence) can be established between them, and vice versa.A similar dependence would be expected in the mset theory, too, when examining two msets with the same total multiplicities.The following elementary example shows that it is not the case.The two msets ⦃⦄ 3 and ⦃, ⦄ 2,1 have the same total multiplicities but obviously, a bijection between the corresponding root sets {} and {, } cannot be found (as it is in the set theory).This means that the notion of total multiplicity in the mset theory is not a generalization of the notion of cardinality in the set theory.Furthermore, if we examine infinite msets, then we will obtain infinite total multiplicities, but this will not allow us to compare these msets in a way analogous to comparing infinite sets.The infinite total multiplicity of the mset may be due to the infinite cardinality of the root sets, or to the infinite multiplicity of some of the elements of the mset, or both.In the case of infinite msets a new concept of cardinality is needed, similar to the concept of the cardinality of infinite sets.For this purpose, we will define the notions of m-injection (one-to-one), m-surjection (onto), m-bijection (one-to-one and onto) and m-cardinality which are applicable to finite and infinite msets, and are a generalization of the corresponding notions of the set theory.On the basis of these notions we will define the binary relations '≺ , ≍, ≻' , through which we will compare the m-cardinality of the finite and infinite msets.
Let  = ⦃ * ,   ⦄ and  = ⦃ * ,   ⦄ be two msets.The modification of the definitions for injection, bijection, and surjection allows for the application of the concept of cardinality to msets in a similar way as in set theory.
For msets  and , we define the binary relations: Definition IV.  ≼  means that there is an m-injection :  →  (or equivalently there exists an m-surjection :  → ).In this case we will say that the m-cardinality of  is less than or equal to the m-cardinality of  and we will denote it in the following way: || ≤ ||.Definition V.  ≍  means that there is an m-bijection :  → .In this case we will say that the m-cardinality of  is equal to the m-cardinality of  and will denote it in the following way: Definition VI.  ≺  means that  ≼  ∧ ∼  ≍ .In this case we will say that the m-cardinality of  is less than the m-cardinality of  and will denote it in the following way: Therefore, the 'm-cardinality' is determined by the total multiplicity of the msets and by the cardinality of the root sets.
By introducing 'm-cardinality', we generalize the definitions of cardinality, injection, bijection, and surjection to apply to msets.
Furthermore, it is not always possible to compare two msets for their m-cardinality.For example, for the two msets ⦃, ⦄ 2,2 and {, , } none of the three relations is fulfilled, i.e., ⦃, ⦄ 2,2 ⊀ {, , } , ⦃, ⦄ 2,2 ⊁ {, , }, ⦃, ⦄ 2,2 ≭ {, , }.The reason for this is that there is neither an m-injection, nor an m-surjection between the two msets.Indeed, |⦃, ⦄ This function is an m-bijection, but not a bijection (only a surjection), see Figure 1.12.If we define the m-cardinality of msets using the relations of injection, bijection, and surjection, then we will arrive at the inequality: On the other hand, if we define the m-cardinality of msets using the relations of m-injection, m-bijection, and m-surjection, then we will arrive at the logical equality This example shows that it is more logical to define the m-cardinality of msets through the relations of m-injection, m-bijection, and m-surjection, rather than through the relations of injection, bijection, and surjection.
Let there be two msets  and , with corresponding root sets  * and  * and corresponding total multiplicities () and ().The following properties follow from the above definitions.

M-cardinality of infinite msets
We will go on to examine infinite msets.With these msets the cardinality of the root set and/or the multiplicity of the individual elements are infinite cardinal numbers.Accordingly, when we operate with these numbers, we will apply the cardinal arithmetic, assuming the Axiom of Choice.
Let us examine the two infinite msets  and ℤ + : Here ℤ + = {1,2,3, . . .} is the set of the positive integers ℤ + and It is clear that ⦃1,2,3, … ⦄ (Here we used formula (15).)Therefore, formula (17) is fulfilled. Theorem 3.7.The following relations hold: Proof.Indeed, Using Property I, This completes the proof. We obtained an infinite number of different sequences of decreasing transfinite m-cardinal numbers which are less than ℵ 0 (the cardinal number corresponding to the countably infinite set).Therefore, if we examine the m-cardinality of infinite msets, it will appear that between the finite numbers and ℵ 0 there are hierarchies of infinities which have not been investigated so far.The following relations hold between msets with negative multiplicity: Finally, we will present a formula that illustrates the relationship between infinite msets and infinite series:

Conclusion
The present work examines and characterizes infinite msets.It presents the possibility of obtaining msets with elements of negative or fraction multiplicity.A new concept of 'm-cardinal number' is introduced, which is a generalization of the concept of cardinal number in the context of multisets.So far it has been assumed that the smallest transfinite cardinal number is ℵ 0 , the cardinal number corresponding to a countably infinite set.Concrete examples of transfinite m-cardinal numbers are given in the paper, corresponding to infinite msets, less than ℵ 0 .
Upon investigating infinite msets, it has been established that there is an infinite number of different sequences of decreasing transfinite m-cardinal numbers which are less than ℵ 0 .Therefore, between the finite numbers and ℵ 0 there are hierarchies of infinities which have not been investigated so far.
Furthermore, there are infinite msets with negative multiplicity which have an m-cardinality less than zero.It has been proved that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, without a smallest transfinite m-cardinal number in this sequence.
The concepts and methods presented can find interesting applications and help the improvement and development of a variety of spheres, such as the set theory, mathematical logic, probability and statistics, combinatorics, order theory, computer science, data science, artificial intelligence, etc.