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

Discrete dynamical systems in group theory

In this expository paper we describe an unifying approach for many known entropies in Mathematics. First we recall the notion of semigroup entropy h_S in the category S of normed semigroups and contractive homomorphisms, recalling also its properties. For a specific category X and a functor F from X to S, we have the entropy h_F, defined by the composition of h_S with F, which automatically satisfies the same properties proved for h_S. This general scheme permits to obtain many of the known entropies as h_F, for appropriately chosen categories X and functors F. In the last part we recall the definition and the fundamental properties of the algebraic entropy for group endomorphisms, noting how its deeper properties depend on the specific setting. Finally we discuss the notion of growth for flows of groups, comparing it with the classical notion of growth for finitely generated groups