Ramsey monoids and aperiodic monoids

One of the most important theorems in the algebraic theory of automata is due to Schützenberger. It states that a rational language is star-free if and only if its syntactic monoid is finite and aperiodic. In Ramsey theory, two very important classes of monoids are those of Ramsey monoids and -controllable monoids. If a monoid belongs to one of these two classes then a coloring statement holds for, the free semigroup of words on. While these notions were formalized recently, they generalize celebrated theorems by Hindman, Carlson, Gowers, Furstenberg, and Katznelson. These notions found applications in set theory, Ramsey spaces, descriptive set theory, and Banach space theory. In joint work with Claudio Agostini, we prove that Ramsey monoids and -controllable monoids are aperiodic and that large classes of finite aperiodic monoids are Ramsey or -controllable. For example, Ramsey monoids are exactly those finite, aperiodic monoids such that is linearly ordered by inclusion. These results seem to show a new connection between Ramsey theory and automata. The aim of this talk is to present our main results to stimulate a dialogue between these two areas.

Authors

• Eugenio Colla

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