Arithmetical categories and commutator theory

Pedicchio, Maria Cristina

doi:10.1007/bf00122258

Cited by 37 publications

(29 citation statements)

References 14 publications

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“…When C is exact Mal'cev, we have the following characterization: Following the varietal case, a category which satisfies the conditions of this theorem is called arithmetical, see [14] and [5]. When C is only regular Mal'cev, the situation given by Theorem 3.1 is characterized by a weak occurrence of the congruence distributivity: THEOREM 3.3.…”

Section: ) Any Internal Groupoid In C Is An Equivalence Relationmentioning

confidence: 95%

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Congruence Distributivity in Goursat and Mal’cev Categories

Bourn

2005

Appl Categor Struct

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We characterize the congruence distributive property for the Goursat and regular Mal'cev categories in terms of preservation of the intersection by direct images. This splits a previous characterization for the exact Mal'cev categories in two cases: plain congruence distributive property and weak congruence distributive property. Mathematics Subject Classifications (2000): Primary: 08B10, 08C05, 18C05; secondary: 08A30, 18D05.

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Section: ) Any Internal Groupoid In C Is An Equivalence Relationmentioning

confidence: 95%

“…This is M. C. Pedicchio who first gave an answer for the exact Mal'cev categories with coequalizers [14]: namely the property that any internal groupoid be an equivalence relation. This first result has been generalized to any exact Mal'cev category in [5].…”

Section: Introductionmentioning

confidence: 98%

Congruence Distributivity in Goursat and Mal’cev Categories

Bourn

2005

Appl Categor Struct

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“…T ∧ R = X and T ∧ S = X ⇒ T ∧ (R ∨ S) = X When the Mal'cev category D is exact, it is stiffly Mal'cev if and only if it is congruence distributive [25]. The aim of this section is to investigate the eccentral and faithful groupoids in this context and, in particular, to show that the dual of any boolean topos E is a groupoid accessible stiffly Mal'cev category in which the only faithful groupoids are the undiscrete equivalence relations ∇ Y .…”

Section: ) Any Internal Groupoid Is An Equivalence Relation;mentioning

confidence: 98%

“…On the other hand, recall that the dual E op of any elementary topos E is an exact stiffly Mal'cev category, see [25] and [7].…”

Section: ) Any Internal Groupoid Is An Equivalence Relation;mentioning

confidence: 99%

Centralizer and faithful groupoid

Bourn

2011

Journal of Algebra

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Communicated by Michel Van den Bergh MSC: primary 20J05, 17B55, 18G50 secondary 18D35, 18D99, 18C99 Keywords: Mal'cev Protomodular and action representative category Split extension classifier Internal groupoid Commutator and centralizerWe correlate existence and aspect of some properties of centralizers with the conceptual notion of Θ-faithful objects. We produce a large choice of examples from standard algebraic situations to topological models, going through less classical cases as the new notions of algebras introduced by Loday (Leibniz algebras, associative dialgebras and trialgebras), or the dual of any boolean topos.

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“…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.…”

Section: Introductionmentioning

confidence: 99%