Ramsey monoids

Recently, Solecki [Forum Math. Sigma 7 (2019), p. 40] introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman’s theorem, Carlson’s theorem, and Gowers’ \operatorname {FIN}_k theorem. He proved that an entire class of finite monoids is Ramsey. Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids. We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey theorem. The results obtained suggest a possible connection with Schützenberger’s theorem and finite automata theory.