Multisets

Abstract

Multisets are sets that allow repetition of elements, therefore accounting for their frequency, or multiplicity of observation. As such, multisets provide flexible resources for scientific modeling. In the present work, after revising the main aspects of traditional sets, we introduce some of the main concepts and characteristics of multisets, which is followed by their generalization to take into account vectors and matrices, and then functions and scalar and vector fields. These developments require multisets to become capable of coping with negative multiplicities, which gives rise to several additional set operations. Then multiset operations can be naturally incorporated into real function spaces allowing, among other possibilities, the definition of a De Morgan theorem between real-valued functions. Special attention is given to understanding the Jaccard and coincidence similarity indices in the context of real-valued multisets and functions, and it is shown that these indices, especially the latter, can yield narrow and sharp peaks corresponding to pattern matchings while attenuating secondary structures.