Universal locally finite maximally homogeneous semigroups and inverse semigroups

In 1959, P. Hall introduced the locally finite group U, today known as Hall’s universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in U. It can be explicitly described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fra¨ıss´e limit of the class of all finite groups. Since its introduction Hall’s group, and several natural generalisations, have been widely studied. In this article we use a generalisation of Fra¨ıss´e theory to construct a countable, universal, locally finite semigroup T , that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup I which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups T and I are the natural counterparts of Hall’s universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall’s group itself

Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups

Let $X$ be a finite set such that $|X|=n$ and let $i\leq j \leq n$. A group G\leq \sym is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.) A group G\leq \sym is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $|I|=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\in G$ such that $Ig$ is a section for $P$. (That is, the orbit of each $k$-subset of $X$ contains a section for each $k$-partition of $X$.) In this paper we classify the groups with the $k$-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the $(k-1,k)$-homogeneous groups (for $2<k\leq \lfloor \frac{n+1}{2}\rfloor$). As a corollary of the classification we prove that a $(k-1,k)$-homogeneous group is also $(k-2,k-1)$-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the $k$-universal transversal property have the $(k-1)$-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank $k$ transformation on $X$ generate a regular semigroup (for $1\leq k\leq \lfloor \frac{n+1}{2}\rfloor$). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.Comment: Includes changes suggested by the referee of the Transactions of the AMS. We gratefully thank the referee for an outstanding report that was very helpful. We also thank Peter M. Neumann for the enlightening conversations at the early stages of this investigatio

Homogeneity and omega-categoricity of semigroups

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