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Coalgebraic methods for Ramsey degrees of unary algebras

Dragan Mašulović

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 $Ω$. We think of unary algebras as $M$-sets where $M = Ω^*$ is the free monoid of words over the alphabet $Ω$ 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 $Ω$ has finite small Ramsey degrees. This generalizes some Ramsey-type results of M.\ Sokić 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 $ω$. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

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 . We think of unary algebras as -sets where is the free monoid of words over the alphabet and show that for an arbitrary monoid (finite or infinite) the class of all finite -sets has finite small Ramsey degrees. This immediately implies that the class of all finite -sets, where 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 has finite small Ramsey degrees. This generalizes some Ramsey-type results of M.\ Sokić concerning finite unary algebras over finite languages and finite -sets for finite groups~. To do so we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat -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 . Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

Paper Structure

This paper contains 8 sections, 21 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chains
  4. Unary algebras, $G$-sets and $M$-sets
  5. Categories and functors
  6. Adjunctions, monads and comonads
  7. Ramsey properties in a category
  8. Ramsey properties of $M$-sets

Key Result

Lemma 2.1

Let $\mathcal{A} = (A, \alpha)$ be an Eilenberg-Moore $E$-coalgebra and $X$ a set. For every mapping $f : A \to X$ there is a unique coalgebra homomorphism $f^\# : \mathcal{A} \to \mathcal{E}(X)$ such that $\varepsilon_X \cdot f^\# = f$.∎

Figures (2)

  • Figure 1: An ordered rooted forest and it representation as an $E$-coalgebra
  • Figure 2: The construction of $f^* : A_\ell \to \omega$

Theorems & Definitions (38)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 2.1
  • Example 4
  • Lemma 3.1
  • Lemma 3.2
  • Theorem \oldthetheorem: The Finite Ramsey Theorem ramsey
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: The Infinite Ramsey Theorem ramsey
  • ...and 28 more