Paper «When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?» published in Log. J. IGPL

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J. Climent Vidal, E. Cosme Llópez.
When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?
Log. J. IGPL, 26(4):381-407, 2018.

doi: 10.1093/jigpal/jzy005

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.

2020 Mathematics Subject Classification: 03C20, 08A68, 18A30.

Keywords: support of a many-sorted set, family of many-sorted algebras with constant support, profinite, retract, projective limit, inductive limit, ultraproduct.