A Study of Multiset Algebras: a Systematization of Fundamentals of Multiset Theory

This dissertation presents a concise history of the uses of multisets in various disguised forms which eventually led to a formalization of multiset theory. We present a comprehensive study of fundamentals of multisets and their applications. We study max?plus algebra and by exploiting the notion of compatibility relation (reflexive and symmetric), we develop a truncated symmetrized max?plus algebra called minimal Smax or ?Smax, and discuss its application. Finally, we outline some multiset algebras developed in the literature. We explicate the notion of multiset space, introduce two new operations on the multiset space and show that it shares properties of many algebraic structures. We also develop the concept of relations and functions on multiset spaces and show that the multiset space can be metrized thereby defining topology on a multiset. In the end, we delineate the problem related to difference and complementation in multiset theory. We show that none of the existing approaches succeeds in resolving the attendant difficulties without assuming some contrived stipulations.