When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?

TL;DR

This paper characterizes when profinite many-sorted algebras are retracts of ultraproducts of finite algebras, providing categorical insights and conditions related to constant support families.

Contribution

It establishes a criterion for profinite many-sorted algebras to be retracts of ultraproducts, with a categorical framework and functorial construction.

Findings

Profinite algebras are retracts of ultraproducts under constant support conditions.

Categorical representation of the retraction property.

Construction of a functorial retraction in a specialized category.

Abstract

For a set of sorts $S$ and an $S$-sorted signature $\Sigma$ we prove that a profinite $\Sigma$-algebra, i.e., a projective limit of a projective system of finite $\Sigma$-algebras, is a retract of an ultraproduct of finite $\Sigma$-algebras if the family consisting of the finite $\Sigma$-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction.