in this paper we investigate the green‎ ‎graphs for the regular and inverse semigroups by considering the‎ ‎green classes of them‎. ‎and by using the properties of these‎ ‎semigroups‎, ‎we prove that all of the five green graphs for the‎ ‎inverse semigroups are isomorphic complete graphs‎, ‎while this‎ ‎doesn't hold for the regular semigroups‎. ‎in other words‎, ‎we prove‎ ‎that in a regular se...

the aim of this paper is to study semigroups possessing $e$-regular elements, where an element $a$ of a semigroup $s$ is {em $e$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ esubseteq e(s)$. where $s$ possesses `enough' (in a precisely defined way) $e$-regular elements, analogues of green's lemmas and even of green's theorem hold, where green's relations $mbox{$...

THE concept of HNN-extensions of groups was introduced by Higman, Neumann and Neumann in 1949. HNN-extensions and amalgamated free products have played a crucial role in combinatorial group theory, especially for algorithmic problems. In inverse semigroup theory there are many ways of constructing HNNextension in order to ensure the embeddability of the original inverse semigroup in the new one...

The aim of this paper is to study semigroups possessing $E$-regular elements, where an element $a$ of a semigroup $S$ is {em $E$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ Esubseteq E(S)$. Where $S$ possesses `enough' (in a precisely defined way) $E$-regular elements, analogues of Green's lemmas and even of Green's theorem hold, where Green's relations ${mathc...

In the structure theory of inverse semigroups, there are two approaches, basically from the 1970’s, to build up inverse semigroups from semilattices and groups via their semidirect products. These approaches are dual to each other in the sense that one produces any inverse semigroup from a semidirect product of a semilattice by a group by taking an idempotent separating homomorphic image of an ...

A class of regular semigroups closed under taking direct products, regular subsemigroups, and homomorphic images is an existence-variety (or evariety) of regular semigroups. Each e-variety of locally inverse semigroups can be characterized by a set of bi-identities. These are identities of terms of type (2, 2) in two sorts of variables X and X'. In this paper we obtain a basis of bi-identities ...

We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup S and a homomorphism φ of S onto a group G, we show, under an assumption on ker(φ), that S has weak containment if and only if G is amenable and ker(φ) has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show tha...

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {·,−1 }, it is also finitely based. It...

As Jonathan Leech has pointed out, many natural examples of inverse semigroups are semilattice ordered under the natural partial order. But there are many interesting examples of semilattice ordered inverse semigroups in which the imposed partial order is not the natural one. In this talk we shall explore the structure and properties of these examples and some other questions related to semilat...