# On ordered hypersemigroups with idempotent ideals, prime and weakly prime ideals

## Authors

• Niovi Kehayopulu Professor Docent Dr. University of Athens

## Keywords:

ordered hypersemigroup, ideal, weakly prime, prime, idempotent, semisimple, left regular, intra-regular

## Abstract

Some well known results on ordered semigroups are examined in case of ordered hypersemigroups. Following the paper in Semigroup Forum 44 (1992), 341--346, we prove the following: The ideals of an ordered hypergroupoid$H$ are idempotent if and only if for any two ideals $A$ and $B$ of $H$, we have $A\cap B=(A*B]$. Let now $H$ be an ordered hypersemigroup. Then, the ideals of $H$ are idempotent if and only if $H$ is semisimple. The ideals of $H$ are weakly prime if and only if they are idempotent and they form a chain. The ideals of $H$ are prime if and only if they form a chain and $H$ is intra-regular. The paper serves as an example to show how we pass from ordered semigroups to ordered hypersemigroups.

2018-01-30

## Section

Algebraic Geometry

## How to Cite

On ordered hypersemigroups with idempotent ideals, prime and weakly prime ideals. (2018). European Journal of Pure and Applied Mathematics, 11(1), 10-22. https://doi.org/10.29020/nybg.ejpam.v11i1.3085

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