Congruence Distributivity in Goursat and Mal’cev Categories

Bourn, Dominique

doi:10.1007/s10485-004-8126-5

Cited by 5 publications

(8 citation statements)

References 13 publications

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“…One of the consequences of one of the main theorems (Theorem 3) is that a regular Mal'tsev category (see [10] and [9]) is congruence distributive if and only if it is a majority category. This result generalizes Pixley's result for varieties, and clarifies a remark of D. Bourn in [7] about whether or not the categories NReg(Top) of topological von Neumann regular rings and BoRg(Top) of topological Boolean rings are fully congruence distributive or not.…”

Section: Theorem 2 ([16]supporting

confidence: 84%

“…where K is the kernel equivalence relation on X associated to f . D. Bourn showed in [7], that a regular Mal'tsev category is congruence distributive if and only if for any regular epimorphism f : X → Y and any equivalence relations α, β ∈ Eq(X), we have f (α ∩ β) = f (α) ∩ f (β) (in fact this was shown more generally for Goursat categories). The proof of the following proposition is essentially the proof of Theorem 2.1 in [7], however we include it for completeness.…”

Section: Pixley's Theorem For Categoriesmentioning

confidence: 93%

“…The proof below is essentially that which can be found in [7], however we include a sketch for completeness. Proof Sketch.…”

Section: Pixley's Theorem For Categoriesmentioning

confidence: 99%

“…For example, in [7], the author remarks that the categories VonReg(Top) of topological Von Neumann regular rings, or BoRg(Top) of topological Boolean rings, are both regular protoarithmetical categories, and therefore they satisfy the above weak version of congruence distributivity. It is then remarked that 'it is far less clear, at this point, if they are fully congruence distributive or not'.…”

Section: Pixley's Theorem For Categoriesmentioning

confidence: 99%

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Characterizations of Majority Categories

Hoefnagel

2019

Appl Categor Struct

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In universal algebra, it is well known that varieties admitting a majority term admit several Mal'tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for regular categories. We prove a categorical version of Bergman's Double-projection Theorem: a regular category is a majority category if and only if every subobject S of a finite product A 1 × A 2 × · · · × A n is uniquely determined by its twofold projections. We also establish a categorical counterpart of the Pairwise Chinese Remainder Theorem for algebras, and characterize regular majority categories by the classical congruence equation α ∩ (β • γ) = (α ∩ β) • (α ∩ γ) due to A.F. Pixley.

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Section: Theorem 2 ([16]supporting

confidence: 84%

Section: Pixley's Theorem For Categoriesmentioning

confidence: 93%

“…The proof below is essentially that which can be found in [7], however we include a sketch for completeness. Proof Sketch.…”

Section: Pixley's Theorem For Categoriesmentioning

confidence: 99%

Section: Pixley's Theorem For Categoriesmentioning

confidence: 99%

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Characterizations of Majority Categories

Hoefnagel

2019

Appl Categor Struct

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“…In [3], the author characterized the congruence distributivity property for regular Goursat categories in terms a preservation of binary meets of equivalence relations by regular-epimorphisms. One of the basic observations is that if f : X → Y is any regular epimorphism in a regular category C and E any equivalence relation on X, then in the notation of Section 4.1 we have…”

Section: Pre-exact Categories and Strict Refinementmentioning

confidence: 99%