Centrality and connectors in Maltsev categories

Abstract

We develop a new approach to the classical property of centrality of equivalence relations. The internal notion of connector allows to clarify classical results in Maltsev varieties and to extend them in the more general context of regular Maltsev categories, hence including the important new examples of Maltsev quasivarieties and of topological Maltsev algebras. We also prove that Maltsev categories can be characterized in terms of a property of connectors.

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

“…So it is a Birkhoff subcategory of C, and then the adjunction (1) in Section 1 gives rise to an admissible Galois structure¯ 1 = ((C, E), (C Ab , Z), F U), as proved in the Theorem 3.5. In order to characterize the coverings relative to¯ 1 the following result will be needed (it extends Proposition 5.3 in [6]): Proposition 4.5 Let C be a regular Mal'cev category such that any regular epimorphism is effective for descent. If R is an equivalence relation on A with [R, ∇ A ] = A , there exists an abelian object Q in C such that R is canonically isomorphic to A × Q.…”

“…Proposition 4.3[6] Let R and S be two equivalence relations on an object X in C, and let f :X → Y be a regular epimorphism in C. If [R, S] = X , then [ f (R), f (S)] = Y .Lemma 4.4Let C be a regular Mal'cev category. Given f : A → B and p : E → B regular epimorphisms in C, let us consider the following pullback:…”

Congruence Distributivity in Goursat and Mal’cev Categories

“…When [R, S] = 0, then R ∩S is abelian. Since C is Mal'cev, when we have R ∩ S = X , then R and S are necessarily connected [7]. p(y, x, z)) makes d 0 a (necessarily abelian) group object in C/X.…”

“…where l R and r S are the sections induced by the maps s 0,R and s 0,S underlying the reflexivity of R and S. Let us recall the following definition, see [7], and also [11]: DEFINITION 3.1. Given any Mal'cev category C, a connector on the pair (R, S) of equivalence relations on an object X in C is a morphism p(x, y, z) which satisfies the identities: p(x, y, y) = x and p(y, y, z) = z.…”

Special reflexive graphs in modular varieties

“…An important aspect of modular varieties is that they admit a good theory of commutators of congruences [7], [10], [11], [16]. Any congruence is an internal reflexive graph, actually a groupoid, and the importance of internal categorical structures in commutator theory has been pointed out in various recents papers [3], [4], [6], [12], [13], [18], [19], [20]. The purpose of this paper is to prove some properties of these internal structures which make it possible to characterize important classes of modular varieties.…”