We compare three approaches to the notion of conjugacy for semigroups, the first one via the transitive closure of the uv ∼ vu relation, the second one via an action of inverse semigroups on themselves by partial transformations, and the third one via characters of finitedimensional representations.

This paper introduces a notion of presentation for locally inverse semigroups and develops graph structure to describe the elements given by these presentations. These graphs will have role similar that Cayley group presentations or Schützenberger monoid However, our considerable differences with latter two, even though generalize both groups semigroups. For example, introduced here are not ‘in...

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups. We follow the terminology of [3, 4, 8]. In this paper all topological spaces are Hausdorff. If S is a semigroup then we denote the subset of idempotents of S by E(S). A topological space S that is algebraically a semigroup ...

A subsemigroup S of an inverse semigroup Q is a left I-order in Q, if every element in Q can be written as a−1b where a, b ∈ S and a−1 is the inverse of a in the sense of inverse semigroup theory. We study a characterisation of semigroups which have a primitive inverse semigroup of left I-quotients.

The Temperley-Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups. This theorem, which generalises a recent result of East about semigroup algebras of inverse semigroups, allows us to easily reproduce the cell...

Self-similar inverse semigroups are defined using automata theory. Adjacency semigroups of s-resolved Markov partitions of Smale spaces are introduced. It is proved that a Smale space can be reconstructed from the adjacency semigroup of its Markov partition, using the notion of the limit solenoid of a contracting self-similar semigroup. The notions of the limit solenoid and a contracting semigr...

We introduce two generalisations of the full symmetric inverse semigroup IX and its dual semigroup I ∗ X – inverse semigroups PI ∗ X and PI ∗ X . Both of them have the same carrier and contain IX . Binary operations on PI∗X and PI ∗ X are reminiscent of the multiplication in IX . We use a convenient geometric way to realise elements from these two semigroups. This enables us to study efficientl...