Finitely Related Semigroups and Associated Structures

An algebra A is said to be finitely related (or has finite degree) if there exists a finite set of finitary relations such that the operations preserving those relations are precisely the term functions of A. Determining when a finite algebra is finitely related is called the finite degree problem. While the finite degree problem for algebras in general is well studied, it is far less developed for semigroups and other semigroup-like structures. In this thesis we will give a partial answer to the finite degree problem for permutative semigroups. We will show that there are large classes of non-finitely related nilpotent monoids and use them to study the preservation and non-preservation of finite relatedness. In the final chapter, we begin the study of the finite degree problem for additively idempotent semirings and pointed semidiscriminator extensions. Here, a tool from the theory of natural dualities will be developed and a link between the finite dualisability of an algebra and the finite relatedness of its pointed semidiscriminator extensions will be established.<p></p>