This paper characterizes directed graphs which are Cayley graphs of strong semilattices of groups and, in particular, strong chains of groups, i.e. of completely regular semigroups which are also called Clifford semigroups. © 2006 Elsevier B.V. All rights reserved.

By a 'representation' we shall mean throughout a representation by n x n matrices with entries from an arbitrary (commutative) field. Clifford has constructed all representations of completely simple semigroups [1; 4]. Munn has determined the representations of finite semigroups for which the corresponding semigroup algebra is semi-simple [6]. It is noted by Clifford and Preston [4] that if S i...

let $s$ be an inverse semigroup and let $e$ be its subsemigroup of idempotents. in this paper we define the $n$-th module cohomology group of banach algebras and show that the first module cohomology group $hh^1_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is zero, for every odd $ninmathbb{n}$. next, for a clifford semigroup $s$ we show that $hh^2_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is a banach space,...

A partial group as defined in [3] is a semigroup S which satisfies the following axioms. (i) For every x ∈ S, there exists a (necessarily unique) element ex ∈ S, called the partial identity of x such that exx =xex =x and if yx =xy =x then ex y = yex = ex. (ii) For every x ∈ S, there exists a (necessarily unique) element x−1 ∈ S, called the partial inverse of x such that xx−1 = x−1x = ex and exx...

Fix a finite semigroup S and let a1, . . . , ak , b be tuples in a direct power S. The subpower membership problem (SMP) asks whether b can be generated by a1, . . . , ak. If S is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in nk. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 o...

This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Säıd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish ...

This is an elementary introduction to the representation theory of finite semigroups. We illustrate Clifford–Munn correspondence between representations a semigroup and its maximal subgroups. The emphasis throughout on naturally occurring examples.