Amenability and geometry of semigroups

We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup

Inverse monoids of partial graph automorphisms

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well

On the Finiteness Problem for Automaton (Semi)groups

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskii, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure