We investigate E
*-dense semi\-groups as analogues of E-dense
semigroups
for semigroups with zero. We give a characterisation theorem for
E
*-dense semigroups whose idempotents form a *-rectangular
band. The construction methods of generalised Rees matrix semigroups
are employed to provide examples and illustrations. Our results are
analogous to those of Weipoltshammer for E-dense semigroups.

E∗-dense E-semigroups

In this paper we study semigroups defined either by means of weak inverses modulo Green’s relation $\mathcal{{H}}$, or by means of the set of completely regular elements. Basic properties of E-inversive semigroups are analyzed, and semigroups whose completely regular elements form a subsemigroup or a rectangular band modulo $\mathcal{{H}}$ are considered. We obtain results that generalize the corresponding results for E-semigroups or semigroups whose idempotents form a rectangular band, and also new results for E-inversive semigroups.

Weak inverses modulo Green’s relation \mathcal{{H}} on E-inversive and group-closed semigroups

This paper is inspired by Mitsch, Petrich, Reilly’s work on E – inversive semigroups and inverse semigroups. We first introduce the notions of a good subdirect product and a good surjective subhomo...

Good subdirect products of E–inversive abundant semigroups

We describe the structure of $E-$dense acts over $E-$dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We consider an application to the discrete log problem in cryptography as well as an application to the same problem using completely regular semigroups.

Actions of $E-$dense semigroups and an application to the discrete log problem

Let S = {a, b, c, ...} and = {, , , ...} be two nonempty sets. S is called a -semigroup if , for all and a, b S and , for all a, b, c S and for all , . An element is said to be an -idempotent for some if = e. A -semigroup S is called an E-inversive -semigroup if for each there exist and such that ax is a -idempotent for some . A -semigroup is called a right E--semigroup if for each -idempotent e and -idempotent f, is a -idempotent. In this paper we investigate different properties of E-inversive -semigroup and right E--semigroup.

E-Inversive Γ-Semigroups

A semigroup S is called E-inversive if for every a ∈ S there is an x ∈ S such that ax is idempotent. The purpose of this paper is the investigation of E-inversive semigroups and semigroups whose idempotents form a subsemigroup. Basic properties are analysed and, in particular, semigroups whose idempotents form a semilattice or a rectangular band are considered. To provide examples and characterizations, the construction methods of generalized Rees matrix semigroups and semidirect products are employed.

Barbara Weipoltshammer

Papers

On Classes of E-Inversive Semigroups and Semigroups Whose Idempotents Form a Subsemigroup