Analysis on Semihypergroups

Abstract

The concepts of semihypergroups and hypergroups were first introduced by C. Dunkl, I. Jewett and R. Spector independently around the year 1972. Till then, a variety of research has been carried out on different areas of hypergroups. However, no extensive study is found so far on the more general category of semihypergroups, which serves as building blocks of the hypergroup theory. In this thesis, we initiate a systematic study of semihypergroups. We introduce and study several natural algebraic and analytic stuctures on semihypergroups, which are well-known in the case of topological semigroups and groups. In particular, we study semihypergroup actions, almost periodic and weakly almost periodic function spaces, ideals, homomorphisms and free structures in the category of semihypergroups, and finally investigate where the theory deviates from the classical theory of (topological) semigroups.