Infinite multisets: Basic properties and cardinality

Abstract

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.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a generalization of cardinal numbers in the context of multisets. Some properties of m-cardinal numbers associated with finite and infinite msets have been researched. Concrete examples of transfinite m-cardinal numbers are given, corresponding to infinite msets which are less than ℵ0 (the cardinality of the countably infinite set). It has been established that between finite numbers and ℵ0 there exist hierarchies of transfinite m-cardinals, corresponding to infinite msets. Furthermore, there are examples of infinite msets with negative multiplicity that have a cardinality less than zero. We prove that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, and in this sequence, there is not a smallest transfinite m-cardinal number.