  Recently, Solecki introduced the notion of Ramsey monoid to produce a common
generalization to theorems such as Hindman's theorem, Carlson's theorem, and
Gowers' 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 Sch\"utzenberger's theorem and finite automata theory.
