# Proaperiodic monoids and model theory

## Sam Van Gool

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

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. From this perspective, the operations of multiplication and ω-power on proaperiodic monoids can be understood in a very concrete way. This point of view allows us to import methods from both topology and model theory, in particular saturated models, into the study of proaperiodic monoids. We use these methods to prove results about ω-terms in the free proaperiodic monoid and well-quasi-orders of factors in related proaperiodic monoids.

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.