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.