On involution $le$-semigroups

Abstract

We deal with involution ordered semigroups possessing a greatest element, we introduce the concepts of $\ast$-regularity, $\ast$-intra-regularity, $\ast$-bi-ideal element and $\ast$-quasi-ideal element in this type of semigroups and, using the right and left ideal elements, we give relations between the regularity and $\ast$-regularity, between intra-regularity and $\ast$-intra-regularity. Finally, we prove that in an involution $\ast$-regular $\vee e$-semigroup every $\ast$-bi-ideal element can be considered as a product of a right and a left ideal element, we describe the form of the filter generated by an element of an involution $\ast$-intra-regular $poe$-semigroup $S$, showing that every $\mathcal{N}$-class of $S$ has a greatest element.