The Reverse Representation Problem

Abstract

Cayley's theorem tells us that all groups $\mathbf{G}$ occur as subgroups of the group of automorphisms over some set $X$. In this paper we consider a `sort-of' converse to this question: given a set $X$ and some transformation group $\mathbf{S}$ over $X$, what are the possible group structures on $X$ that result in groups represented by $\mathbf{S}$? 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 the structure of a heap. We study this phenomenon in depth and then move onto looking at particular classes of semigroups namely monoids, groups, inverse semigroups and Clifford semigroups.