Loading...
5 results
Search Results
Now showing 1 - 5 of 5
- On difunctionality of class relationsPublication . Hoefnagel, Michael; Janelidze, Zurab; Rodelo, DianaFor a given variety V of algebras, we define a class relation to be a binary relation R subset of S(2)which is of the form R = S-2 boolean AND K for some congruence class K on A(2), where A is an algebra in V such that S subset of A. In this paper we study the following property of V : every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal'tsev condition on the variety and in a suitable sense, it is a join of Chajda's egg-box property as well as Duda's direct decomposability of congruence classes.
- Facets of congruence distributivity in Goursat categoriesPublication . Gran, Marino; Rodelo, Diana; Nguefeu, Idriss TchoffoWe give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations. (C) 2020 Elsevier B.V. All rights reserved.
- Intrinsic Schreier split extensionsPublication . Montoli, Andrea; Rodelo, Diana; Van der Linden, TimIn the context of regular unital categories we introduce an intrinsic version of the notion of a Schreier split epimorphism, originally considered for monoids. We show that such split epimorphisms satisfy the same homological properties as Schreier split epimorphisms of monoids do. This gives rise to new examples of S-protomodular categories, and allows us to better understand the homological behaviour of monoids from a categorical perspective.
- On lax protomodularity for Ord-enriched categoriesPublication . Clementino, Maria Manuel; Montoli, Andrea; Rodelo, DianaOur main focus concerns a possible lax version of the algebraic property of protomodularity for Ord -enriched categories. Having in mind the role of comma objects in the enriched context, we consider some of the characteristic properties of protomodularity with respect to comma objects instead of pullbacks. We show that the equivalence between protomodularity and certain properties on pullbacks also holds when replacing conveniently pullbacks by comma objects in any finitely complete category enriched in Ord, and propose to call lax protomodular such Ord -enriched categories. We conclude by studying this sort of lax protomodularity for the category OrdAb of preordered abelian groups, equipped with a suitable Ord -enrichment, and show that OrdAb fulfills the equivalent lax protomodular properties with respect to the weaker notion of precomma object; we call such categories lax preprotomodular. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).
- A note on Mal’tsev objectsPublication . Clementino, Maria Manuel; Rodelo, DianaThe aim of this work is to compare the distinct notions of Mal'tsev object in the sense of Weighill and in the sense of Montoli-Rodelo-Van der Linden.