In this thesis we study problems in the theory of semigroups which arise from model theoretic notions. Our focus will be on omega-categoricity and homogeneity of semigroups, a common feature of both of these properties being symmetricity. A structure is homogeneous if every local symmetry can be extended to a global symmetry, and as such it will have a rich automorphism group.
On the other hand, the Ryll-Nardzewski Theorem dictates that omega-categorical structures have oligomorphic automorphism groups.
Numerous authors have investigated the homogeneity and omega-categoricity of algebras including groups, rings, and of relational structures such as graphs and posets. The central aim of this thesis is to forge a new path through the model theory of semigroups.
The main body of this thesis is split into two parts. The first is an exploration into omega-categoricity of semigroups. We follow the usual semigroup theoretic method of analysing Green's relations on an omega-categorical semigroup, and prove a finiteness condition on their classes. This work motivates a generalization of characteristic subsemigroups, and subsemigroups of this form are shown to inherit omega-categoricity. We also explore methods for building omega-categorical semigroups from given omega-categorical structures.
In the second part we study the homogeneity of certain classes of semigroups, with an emphasis on completely regular semigroups. A complete description of all homogeneous bands is achieved, which shows them to be regular bands with homogeneous structure semilattices. We also obtain a partial classification of homogeneous inverse semigroups. A complete description can be given in a number of cases, including inverse semigroups with finite maximal subgroups, and periodic commutative inverse semigroups. These results extend the classification of homogeneous semilattices by Droste, Truss, and Kuske. We pose a number of open problems, that we believe will open up a rich subsequent stream of research.
