We introduce strong Morita equivalence for inverse semigroups. This notion encompasses Mark Lawson’s concept of enlargement. Strongly Morita equivalent inverse semigroups have Morita equivalent universal groupoids in the sense of Paterson and hence strongly Morita equivalent universal and reduced C∗-algebras. As a consequence we obtain a new proof of a result of Khoshkam and Skandalis showing t...

Let S be a semigroup. The elements a, b ∈ S are called primarily conjugate if a = xy and b = yx for certain x, y ∈ S. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case when S is a monoid, denote by G the group of units of S. Then the relation of G-conjugacy is defined by a ∼G b ⇐⇒ a = gbg for certain g ∈ G. We establish the structur...

We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.

in this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric connes amenability. we determine symmetric module amenability and symmetric connes amenability of some concrete banach algebras. indeed, it is shown that $ell^1(s)$ is  a symmetric $ell^1(e)$-module amenable if and only if $s$ is amenable, where $s$ is an inverse semigroup with subsemigr...

There has been much work done recently on the action of semigroups on sets with some important applications to, for example, the theory and structure of semigroup amalgams. It seems natural to consider the actions of semigroups on sets ‘with structure’ and in particular on graphs and trees. The theory of group actions has proved a powerful tool in combinatorial group theory and it is reasonable...

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely g...

We obtain Schur-Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations. AMS Subject Classification: 20M18; 16S99; 20M30; 05E10

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoidIX , that is, a monoid of partial one-to-one selfmaps of a set X . The present paper describes the structure of a categorical dual I Ł X to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symme...

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...