On Reverse Representation
Date
2024-12
Authors
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Publisher
Stellenbosch University
Abstract
Arthur Cayley’s famous representation theorem demonstrates to us that every group G can be realized as a subgroup of the group of automorphisms over some set X. Building on this foundational theorem, one can easily prove similar results for monoids and most generally faithful semigroups. The class of all semigroups with the property that ∀x,y∈X∀z∈X [xz = yz ⇒ x = y]. In this thesis, I establish the concept of reverse Cayley representation; a method of associating with any faithful semigroup S of transformations of a set X a (possibly empty) set of semigroups called unrepresentations. The set of unrepresentations of S is the set of exactly those faithful semigroups X on X with the property that X is represented as S. The goal of this thesis is to delve into reverse representation by defining and characterizing the unrepresentations of transformation structures at various levels of generality. We will look at a number of structures from the previously mentioned faithful semigroups of transformations to transformation groups. At each level I explore the fashion in which these unrepresentations arise and connect them to previously existing, or new, concepts. As I explore these unrepresentations, I will demonstrate that the set of unrepresentations of S is not merely a collection of structures, but has an algebraic structure itself. It is a heap, and this heap of unrepresentations is closely related to the group of ‘internal symmetries’ of any unrepresentation of S. This insight enables us to extend the concept of an invertible element into the realm of semigroups. In addition to this foundational work, I also investigate two special cases of unrepresentation: In the context of Clifford semigroups, Is how h ow t he unrepresentation of a structure can be broken up into various ‘smaller’ unrepresentations of component structures (along with an condition which specifies how these smaller unrepresentations should interact). Furthermore, in the context of category theory, I illustrate how the concept of representation (and thus unrepresentation) can be generalized- not from a group to a semigroup, but from a set to a graph.