Fixed Points and Continuity of Semihypergroup Actions

TL;DR

This paper studies continuous actions on semihypergroups, analyzing fixed-point properties and their relation to invariant means, advancing the understanding of the algebraic and analytic structure of semihypergroups.

Contribution

It introduces and examines continuous actions on semihypergroups and explores their fixed-point properties and connections to invariant means, providing new insights into their structure.

Findings

Continuity properties of actions on semihypergroups are characterized.

Fixed-point properties are linked to the existence of left-invariant means.

The paper establishes equivalences between different fixed-point conditions.

Abstract

In a couple of previous papers, we initiated a systematic study of semihypergroups and had a thorough discussion on certain analytic and algebraic aspects associated to this class of objects. In this article, we introduce and examine (separately) continuous actions on the category of semihypergroups. In particular, we discuss the continuity properties of such actions and explore the equivalence relations between different fixed-point properties of certain actions and the existence of left-invariant mean(s) on the space of almost periodic functions on a semihypergroup.