An Application of le-semigroup Techniques to Semigroups, Γ-semigroups and to Hypersemigroups

Abstract

An $le$-semigroup, is a semigroup $S$ at the same time a lattice with a greatest element $e$ ($e\ge a$ for every $a\in S$) such that $a(b\vee c)=ab\vee ac$ and $(a\vee b)c=ac\vee bc$ for all $a,b,c\in S$. If $S$ is not a lattice but only an upper semilattice ($\vee$-semilattice), then is called $\vee e$-semigroup. A $poe$-semigroup is a semigroup $S$ at the same time an ordered set with a greatest element $e$ such that $a\le b$ implies $ac\le bc$ and $ca\le cb$ for all $c\in S$. Every $\vee e$-semigroup is a $poe$-semigroup. If $S$ is a semigroup or a $\Gamma$-semigroup, then the set ${\cal P}(S)$ of all subsets of $S$ is an $le$-semigroup. If $S$ is an hypersemigroup, then the set ${\cal P^*}(S)$ of all nonempty subsets of $S$ is an $le$-semigroup. So all the results of $le$-semigroups, $\vee e$-semigroups and $poe$-semigroups based on ideal elements, automatically hold for semigroups, $\Gamma$-semigroups and hypersemigroups. This is not the case for  ordered $\Gamma$-semigroups or ordered hypersemigroups; however the main idea, even in these cases, comes from the $le$ ($\vee e$)-semigroups. As an example, we study the weakly prime ideal elements of a $\vee e$-semigroup and their role to the different type of semigroups mentioned above.