On hypersemigroups

TL;DR

This paper characterizes bi-ideals in regular hypersemigroups and establishes conditions for regularity involving idempotent ideals and quasi-ideals, providing foundational insights into hypersemigroup structure.

Contribution

It offers a characterization of bi-ideals and regular hypersemigroups, linking ideal properties and hypersemigroup regularity, and clarifies the conceptual understanding of hypersemigroups.

Findings

Bi-ideals are characterized as products of right and left ideals.

A hypersemigroup is regular iff its ideals are idempotent.

Product of right and left ideals forms a quasi-ideal.

Abstract

We prove that a nonempty subset $B$ of a regular hypersemigroup $H$ is a bi-ideal of $H$ if and only if it is represented in the form $B=A*C$ where $A$ is a right ideal and $C$ a left ideal of $H$. We also show that an hypersemigroup $H$ is regular if and only if the right and the left ideals of $H$ are idempotent, and for every right ideal $A$ and every left ideal $B$ of $H$, the product $A*B$ is a quasi-ideal of $H$. Our aim is not just to add a publication on hypersemigroups but, mainly, to publish a paper which serves as an example to show what an hypersemigroup is and give the right information concerning this structure. We never work directly on an hypersemigroup. If we want to get a result on an hypersemigroup, then we have to prove it first for a semigroup and transfer its proof to hypersemigroup. But there is further interesting information concerning this structure as well, we will deal with at another time.