The Reverse Representation Problem

TL;DR

This paper explores the inverse problem of Cayley's theorem, characterizing the group structures on a set that can be represented by a given transformation group, revealing solutions form a heap structure and analyzing specific semigroup classes.

Contribution

It introduces the concept of unrepresentations as the inverse of Cayley's theorem, characterizes their heap structure, and extends analysis to various semigroup classes.

Findings

Solutions form a heap structure

Characterization of unrepresentations for monoids and groups

Analysis of inverse and Clifford semigroups

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.