Abstract Cayley’s theorem tells us that all groups $$\textbf{G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> occur as subgroups of the group of permutations over some set X . In this paper we consider a ‘sort-of’ converse to this question: given a set X and some transformation group $$\textbf{S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> over X , what are the possible group structures on X that result in groups represented by $$\textbf{S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> ? We solve this problem in the more general setting of faithful semigroups and observe that the solutions to this problem, which we term unrepresentations , have an inherent group structure. We study this phenomenon in depth before finishing with an analysis of the special case of unrepresentations of Clifford semigroups.

Abstract An element e of an ordered semigroup $$\left( S,\cdot ,\le \right) $$ S , · , ≤ is called idempotent (resp. generalised idempotent) if $$e\le {{e}^{2}}$$ e ≤ e 2 (resp. $$\left( e,{{e}^{2}} \right) \in {{\Re }_{\le }}$$ e , e 2 ∈ ℜ ≤ where $${{\Re }_{\le }}$$ ℜ ≤ is the smallest congruence on S containing the relation $$\le \cup {{\le }^{-1}}$$ ≤ ∪ ≤ - 1 ). The set of all idempotents (resp. generalised idempotents) of S is denoted by $$E\left( S \right) $$ E S (resp. $${{E}^{G}}\left( S \right) $$ E G S ). S is called orthodox if (i) the set $$E\left( S \right) $$ E S is non empty and (ii) $$ef\in {{E}^{G}}\left( S \right) $$ e f ∈ E G S for every $$e,f\in E\left( S \right) $$ e , f ∈ E S . An element x in S is an inverse (resp. generalised inverse) of an element a of S if $$a\le axa$$ a ≤ a x a and $$x\le xax$$ x ≤ x a x (resp. $$\left( a,axa \right) ,\left( x,xax \right) \in {{\Re }_{\le }}$$ a , a x a , x , x a x ∈ ℜ ≤ ). We study the notions of generalised inverse and generalised idempotent element and we show that, in an orthodox ordered semigroup, if we know a single generalised inverse of an element a , then we know the set of all generalised inverses of a . We also study the structure of orthodox ordered semigroups giving basic properties of orthodox ordered semigroups and equivalent conditions (based on inverse, generalised inverse, idempotent, generalised idempotent elements) according to which an ordered semigroup is orthodox.

Abstract We describe the structure of E -dense acts over E -dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We also study cancellative actions of semigroups as a type of generalisation of group actions and characterise locally free cancellative acts over E -dense semigroups that are also E -unitary.