# Studying profinite monoids via logic

## Sam Van Gool

### CCNY (CUNY) & ILLC (University of Amsterdam)

This talk is about my ongoing joint research project with Benjamin Steinberg (CCNY). We begin with the observation that the free profinite aperiodic monoid over a finite set *A* is isomorphic to the Stone dual space (spectrum) of the Boolean algebra of first-order definable sets of finite *A*-labelled linear orders (“*A*-words”). This means that elements of this monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite *A*-words. We exploit this view of the free profinite aperiodic monoid to prove both old and new things about it using methods from model theory, in particular (weakly) saturated models.

The talk is aimed at anyone with a basic knowledge of model theory, not necessarily of profinite monoids; in particular I will take care to review some background on profinite monoids and on how they relate to logic and regular languages.

Sam Van Gool is a Marie Skłodowska-Curie post-doctoral fellow, currently working in the Mathematics Department of the City College of New York. His research interests are duality theory and its applications to mathematical logic and computer science, including automata, semigroups, and modal logics.