# Blog Archives

# Topic Archive: monoids

# Proaperiodic monoids and model theory

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.

# Studying profinite monoids via logic

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.