Dual Ramsey properties for classes of algebras

TL;DR

This paper demonstrates that all nontrivial varieties of finite algebras possess dual Ramsey properties, establishing finite dual small and big Ramsey degrees for these classes using novel categorical strategies.

Contribution

It introduces new methods based on adjoint functors and monad categories to prove dual Ramsey properties for classes of finite algebras, a contrast to the abundance of classes with the classical Ramsey property.

Findings

Finite algebras in any nontrivial variety have finite dual small Ramsey degrees.

Every finite algebra has finite dual big Ramsey degree in the free algebra on countably many generators.

Ordered versions of the statements support these dual Ramsey properties.

Abstract

Almost any reasonable class of finite relational structures has the Ramsey property or a precompact Ramsey expansion. In contrast to that, the list of classes of finite algebras with the precompact Ramsey expansion is surprisingly short. In this paper we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various \emph{dual} Ramsey properties. We develop a completely new set of strategies that rely on the fact that left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a monad. We 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.