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# 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 $\varSigma$ of equations and a set X of variables, let $F_{\varSigma }(X)$ be the free $\varSigma -$algebra over X and $\mathcal {V}(\varSigma )$ the variety of all algebras satisfying $\varSigma .$ We consider the question, under which conditions the Set-functor $F_{\varSigma }$ 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 $\varSigma '$ of a set of equations $\varSigma$ as defined by Dent, Kearnes and Szendrei in [3], we show that a functor $F_{\varSigma }$ (weakly) preserves preimages if and only if $\varSigma$ implies its own derivative, i.e. $\varSigma \vdash \varSigma '$, which amounts to saying that weak independence implies independence for each variable occurrence in a term of $\mathcal {V}(\varSigma )$. As a corollary, we obtain that the free-algebra functor will never preserve preimages when $\mathcal {V}(\varSigma )$ is congruence modular.Regarding preservation of kernel pairs, we show that for n-permutable varieties $\mathcal {V}(\varSigma ),$ the functor $F_{\varSigma }$ weakly preserves kernel pairs if and only if $\mathcal {V}(\varSigma )$ is a Mal’cev variety, i.e. 2-permutable.
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Submitted on : Friday, May 21, 2021 - 2:57:47 PM
Last modification on : Wednesday, March 9, 2022 - 3:12:02 PM
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H. 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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