Metric spaces of multisets (Q1363261)

A multiset (MS), or a set with repetitions, is a set of elements \(X=\{x_i\}\) in which the same element can occur several times. The multiplicity of occurrence of an element is essential for a MS. The concept of a multiset (bag) is known in combinatorial mathematics; it is used in programming, in Petri net theory, and in other applications. We introduce in this work some new notions and operations in the field of MS, in particular: families of MS, measure of a MS, and metric spaces of MS. We formulate necessary and sufficient conditions of the existence of various metrics and investigate some of their properties.