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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
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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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