On Semilattice Congruences on Hypersemigroups and on Ordered Hypersemigroups

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 $\mathcal{A}$ of proper prime ideals of $H$ such that $\sigma$ is the intersection of the semilattice congruences ${\sigma }_I$, $I\in \mathcal{A}$ (${\sigma }_I$ is the known relation defined by $a{\sigma }_{I}b$ $\iff$ $a,b\in I$ 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 on  $(S,\cdot ,\le )$, then it is a complete semilattice congruence on $(S,\circ ,\le )$. Illustrative examples are given.