# On Semilattice Congruences on Hypersemigroups and on Ordered Hypersemigroups

## Authors

• Niovi Kehayopulu Professor Docent Dr. University of Athens, Department of Mathematics

## Keywords:

hypergroupoid, ordered hypersemigroup, semilattice congruence, complete (pseudocomplete) semilattice congruence, filter, prime ideal

## Abstract

We prove that if $H$ is an hypersemigroup (resp. ordered hypersemigroup) and $\sigma$ is a semilattice congruence (resp. complete semilattice congruence) on $H$, then there exists a family $\cal A$ of proper prime ideals of $H$ such that $\sigma$ is the intersection of the semilattice congruences $\sigma_I$, $I\in\cal A$ ($\sigma_I$ is the known relation defined by $a\sigma_I b$ $\Leftrightarrow$ $a,b\in I$ or $a,b\notin I$). Furthermore, we study the relation between the semilattices of an ordered semigroup and the ordered hypersemigroup derived by the hyperoperations $a\circ b=\{ab\}$ and $a\circ b:=\{t\in S \mid t\le ab\}$. We introduce the concept of a pseudocomplete semilattice congruence as a semilattice congruence $\sigma$ for which $\le\subseteq\sigma$ and we prove, among others, that if $(S,\cdot,\le)$ is an ordered semigroup, $(S,\circ,\le)$ the hypersemigroup defined by $t\in a\circ b$ if and only if $t\le ab$ and $\sigma$ is a pseudocomplete semilattice congruence onÂ  $(S,\cdot,\le)$, then it is a complete semilattice congruence on $(S,\circ,\le)$. Illustrative examples are given.

2018-04-27

## Section

Algebraic Topology

## How to Cite

On Semilattice Congruences on Hypersemigroups and on Ordered Hypersemigroups. (2018). European Journal of Pure and Applied Mathematics, 11(2), 476-492. https://doi.org/10.29020/nybg.ejpam.v11i2.3266