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&apos...

Recent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting ...

We extend the ‘∨-premorphisms’ part of the Ehresmann-ScheinNambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type t...

We prove a structure result on proper extensions of two-sided restriction semigroups in terms partial actions, generalizing respective results for monoids and inverse upgrading the latter. introduce study several classes actions that generalize semigroups. establish an adjunction between category P(S) semigroup S A(S) subject to certain conditions going back work O'Carroll. In A(S), we specify ...

A factorizable inverse monoid can be identified, up to isomorphism, with an inverse submonoid M of a symmetric inverse monoid I(X) where each element of M is a restriction of a permutation of X belonging to M . So factorizable inverse monoids are natural objects, and appear in a number of branches of mathematics, cf. [12], [4]. The notion of an almost factorizable inverse semigroup was introduc...

Abstract. Inductive constellations are one-sided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we define the notions of the action and partial action of an inductive constellation on a set, before introducing the Szendrei expansion of an inductive co...