Categorical equivalence and the Ramsey property for finite powers of a primal algebra

Abstract

In this paper, we investigate the best known and most important example of a categorical equivalence in algebra, that between the variety of boolean algebras and any variety generated by a single primal algebra. We consider this equivalence in the context of Kechris-Pestov-Todorčević correspondence, a surprising correspondence between model theory, combinatorics and topological dynamics. We show that relevant combinatorial properties (such as the amalgamation property, Ramsey property and ordering property) carry over from a category to an equivalent category. We then use these results to show that the category whose objects are isomorphic copies of finite powers of a primal algebra ${\mathcal{A}}$ together with a particular linear ordering <, and whose morphisms are embeddings, is a Ramsey age (and hence a Fraïssé age). By the Kechris-Pestov-Todorčević correspondence, we then infer that the automorphism group of its Fraïssé limit is extremely amenable. This correspondence also enables us to compute the universal minimal flow of the Fraïssé limit of the class ${{\bf V}_{fin} \mathcal{(A)}}$ whose objects are isomorphic copies of finite powers of a primal algebra ${\mathcal{A}}$ and whose morphisms are embeddings.