A syntactic characterization of weakly Mal’tsev varieties. (English) Zbl 07903761
Summary: The notion of a weakly Mal’tsev category, as it was introduced in [ibid. 21, 91–117 (2008; Zbl 1166.18005)] by the third author, is a generalization of the classical notion of a Mal’tsev category. It is well-known that a variety of universal algebras is a Mal’tsev category if and only if its theory admits a Mal’tsev term. In the main theorem of this paper, we prove a syntactic characterization of the varieties that are weakly Mal’tsev categories. We apply our result to the variety of distributive lattices which was known to be a weakly Mal’tsev category before. By a result of Z. Janelidze and the third author [ibid. 27, 65–79 (2012; Zbl 1252.18008)], a finitely complete category is weakly Mal’tsev if and only if any internal strong reflexive relation is an equivalence relation. In the last part of this paper, we give a syntactic characterization of those varieties in which any regular reflexive relation is an equivalence relation.
MSC:
| 08B05 | Equational logic, Mal’tsev conditions |
| 18E13 | Protomodular categories, semi-abelian categories, Mal’tsev categories |
| 18C10 | Theories (e.g., algebraic theories), structure, and semantics |
| 06B20 | Varieties of lattices |
| 06D99 | Distributive lattices |
| 18D40 | Internal categories and groupoids |
Keywords:
weakly Mal’tsev category; weakly Mal’tsev variety; Mal’tsev condition; syntactic characterization; strong relation; pullback injectionReferences:
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| [28] | Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium Department of Mathematics, Royal Military Academy, Brussels, Belgium Instituto Politécnico de Leiria, Leiria, Portugal Email: adja.egner@uclouvain.be pierre-alain.jacqmin@uclouvain.be martins.ferreira@ipleiria.pt This article may be accessed at http://www.tac.mta.ca/tac/ THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. Full text of the journal is freely available at http://www.tac.mta.ca/tac/. |
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| [31] | Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Julie Bergner, University of Virginia: jeb2md (at) virginia.edu Richard Blute, Université d’ Ottawa: rblute@uottawa.ca John Bourke, Masaryk University: bourkej@math.muni.cz Maria Manuel Clementino, Universidade de Coimbra: mmc@mat.uc.pt Valeria de Paiva, Topos Institute: valeria.depaiva@gmail.com Richard Garner, Macquarie University: richard.garner@mq.edu.au |
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