Victoria Gould (University of York)

Pseudo-finite semigroups and diameter

A semigroup $S$ is said to be (right) pseudo-finite if the universal right congruence $S \times S$ can be generated by a finite set $U$ of pairs of elements of $S$ and there is a bound on the length of derivations for an arbitrary pair as a consequence of those in $U$ . The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets.

The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Zelazko for Banach algebras. The property also arises from several other sources.

Without assuming any prior knowledge, this talk investigates the somewhat unpredictable notion of pseudo-finiteness. Some well-known uncountable semigroups have diameter $1$; on the other hand, a pseudo-finite group is forced to be finite. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools all play a part.

This research sits in the wider framework of a study of finitary conditions for semigroups.