Conference Papers Year : 2020

Free-Algebra Functors from a Coalgebraic Perspective

Abstract

We continue our study of free-algebra functors from a coalgebraic perspective as begun in [8]. Given a set Σ

of equations and a set X of variables, let FΣ(X)
be the free Σ
algebra over X and V(Σ)
the variety of all algebras satisfying Σ.
We consider the question, under which conditions the Set-functor FΣ
weakly preserves pullbacks, kernel pairs, or preimages [9].We first generalize a joint result with our former student Ch. Henkel, asserting that an arbitrary Set
endofunctor F weakly preserves kernel pairs if and only if it weakly preserves pullbacks of epis.By slightly extending the notion of derivative Σ
of a set of equations Σ
as defined by Dent, Kearnes and Szendrei in [3], we show that a functor FΣ
(weakly) preserves preimages if and only if Σ
implies its own derivative, i.e. ΣΣ
, which amounts to saying that weak independence implies independence for each variable occurrence in a term of V(Σ)
. As a corollary, we obtain that the free-algebra functor will never preserve preimages when V(Σ)
is congruence modular.Regarding preservation of kernel pairs, we show that for n-permutable varieties V(Σ),
the functor FΣ
weakly preserves kernel pairs if and only if V(Σ)
is a Mal’cev variety, i.e. 2-permutable.

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Dates and versions

hal-03232349 , version 1 (21-05-2021)

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H. Peter Gumm. Free-Algebra Functors from a Coalgebraic Perspective. 15th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2020, Dublin, Ireland. pp.55-67, ⟨10.1007/978-3-030-57201-3_4⟩. ⟨hal-03232349⟩
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