Dual Ramsey properties for classes of algebras

Abstract

Almost any reasonable class of finite relational structures has the Ramsey property or a Ramsey expansion. In contrast to that, the list of classes of finite algebras with the Ramsey expansion is surprisingly short. In this paper we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that right adjoints preserve the Ramsey property while left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a (co)monad. We first show that for any group G (finite or infinite) finite G-sets have finite small Ramsey degrees, and that every finite G-set has a finite big Ramsey degree in the cofree G-set on countably many cofree generators. We then show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements. To the best of our knowledge, this is the first calculation of dual big Ramsey degrees after the Infinite Dual Ramsey Theorem of Carlson and Simpson.

Coalgebraic methods for Ramsey degrees of unary algebras

“…Lemma 3.2. [13] Let C be a locally small category such that all the morphisms in C are mono. Let A be a full subcategory of C, let A, B, D ∈ Ob(A) and The Ramsey property for ordered structures implies the existence of finite small Ramsey degrees for the corresponding unordered structures.…”

“…The proof of the dual version of the theorem below is given in [13]. Just as an illustration we provide the proof of the "direct" version here.…”

“…11. (cf [13,. Theorem 3.15]) Let C be a locally small category, E : C → C a functor and δ : E → EE a comultiplication for E. If C has the Ramsey property then so does every full subcategory of EM w (E, δ) which contains all the cofree Eilenberg-Moore E-coalgebras.Proof.…”