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\xa0 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.