Analysis on Semihypergroups: Function Spaces, Homomorphisms and Ideals

TL;DR

This paper systematically studies semihypergroups by exploring their algebraic and analytic structures, including function spaces, homomorphisms, and ideals, to understand their properties and deviations from classical semigroup theory.

Contribution

It introduces new structures and analyzes properties of semihypergroups, extending classical concepts from topological groups and semigroups.

Findings

Characterization of almost periodic function spaces on semihypergroups

Analysis of homomorphisms and ideals in semihypergroups

Insights into the structure and deviations from classical semigroup theory

Abstract

The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by C. Dunkl [4], I. Jewett [13] and R. Spector [20] independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups.