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

Abstract

For a set of sorts $S$ and an $S$-sorted signature $Σ$ we prove that a profinite $Σ$-algebra, i.e., a projective limit of a projective system of finite $Σ$-algebras, is a retract of an ultraproduct of finite $Σ$-algebras if the family consisting of the finite $Σ$-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.