Coalgebraic methods for Ramsey degrees of unary algebras

Abstract: In this paper we are interested in the existence of small and big Ramsey degrees of classes of finite unary algebras in arbitrary (not necessarily finite) algebraic language $\Omega$. We think of unary algebras as $M$-sets where $M = \Omega^*$ is the free monoid of words over the alphabet $\Omega$ and show that for an arbitrary monoid $M$ (finite or infinite) the class of all finite $M$-sets has finite small Ramsey degrees. This immediately implies that the class of all finite $G$-sets, where $G$ is an arbitrary group (finite or infinite), has finite small Ramsey degrees, and that the class of all finite unary algebras over an arbitrary (finite or infinite) algebraic language $\Omega$ has finite small Ramsey degrees. This generalizes some Ramsey-type results of M.\ Soki\'c concerning finite unary algebras over finite languages and finite $G$-sets for finite groups~$G$. To do so we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat $M$-sets as Eilenberg-Moore coalgebras for "half a comonad" and using pre-adjunctions transport the Ramsey properties we are interested in from the category of finite or countably infinite chains of order type $\omega$. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.