On involution $le$-semigroups

TL;DR

This paper explores involution ordered semigroups with a greatest element, introducing new regularity concepts and analyzing their structural properties, including the behavior of bi-ideal elements and the structure of generated filters.

Contribution

It introduces the concepts of $*$-regularity, $*$-intra-regularity, and analyzes their relations with ideal elements in involution $le$-semigroups, providing new structural insights.

Findings

In involution $*$-regular $igvee e$-semigroups, every $*$-bi-ideal element is a product of right and left ideal elements.

The filter generated by an element in involution $*$-intra-regular $poe$-semigroups has a greatest element.

Every $ ext{ extcal N}$-class in such semigroups contains a greatest element.

Abstract

We deal with involution ordered semigroups possessing a greatest element, we introduce the concepts of $*$-regularity, $*$-intra-regularity, $*$-bi-ideal element and $*$-quasi-ideal element in this type of semigroups and, using the right and left ideal elements, we give relations between the regularity and $*$-regularity, between intra-regularity and $*$-intra-regularity. Finally, we prove that in an involution $*$-regular $\vee e$-semigroup every $*$-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 $*$-intra-regular $poe$-semigroup $S$, showing that every $\cal N$-class of $S$ has a greatest element.