Finite and profinite aperiodic monoids have long played a fundamental role in automata and semigroup theory. The famous Schützenberger theorem shows that the finite aperiodic monoids recognize precisely the star-free languages; this class was later shown to be the class of first-order definable languages, thus bringing logic into the picture.

From the perspective of model theory, these results say that elements of the free pro-aperiodic monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite A-words. This makes it possible to understand the operations of multiplication and omega-power in a very concrete way. We thus import methods from both topology and model theory, in particular saturated models, into the study of pro-aperiodic monoids.

This talk is on joint work with Ben Steinberg: https://arxiv.org/pdf/1609.07736v1.pdf