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

摘要

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-TodoreviAc 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 together with a particular linear ordering <, and whose morphisms are embeddings, is a Ramsey age (and hence a Fra < ss, age). By the Kechris-Pestov-TodoreviAc correspondence, we then infer that the automorphism group of its Fra < ss, limit is extremely amenable. This correspondence also enables us to compute the universal minimal flow of the Fra < ss, limit of the class whose objects are isomorphic copies of finite powers of a primal algebra and whose morphisms are embeddings.