%0 Conference Proceedings
%T Free-Algebra Functors from a Coalgebraic Perspective
%+ Philipps Universität Marburg = Philipps University of Marburg
%A Gumm, H., Peter
%< avec comité de lecture
%( Lecture Notes in Computer Science
%B 15th International Workshop on Coalgebraic Methods in Computer Science (CMCS)
%C Dublin, Ireland
%Y Daniela Petrişan
%Y Jurriaan Rot
%I Springer International Publishing
%3 Coalgebraic Methods in Computer Science
%V LNCS-12094
%P 55-67
%8 2020-04-25
%D 2020
%R 10.1007/978-3-030-57201-3_4
%Z Computer Science [cs]Conference papers
%X 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.
%G English
%Z TC 1
%Z WG 1.3
%2 https://inria.hal.science/hal-03232349/document
%2 https://inria.hal.science/hal-03232349/file/493577_1_En_4_Chapter.pdf
%L hal-03232349
%U https://inria.hal.science/hal-03232349
%~ IFIP-LNCS
%~ IFIP
%~ IFIP-TC
%~ IFIP-TC1
%~ IFIP-WG1-3
%~ IFIP-CMCS
%~ IFIP-LNCS-12094