Characterizations of Majority Categories

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 Do...

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Bibliographic Details
Published in:Applied categorical structures 2020-02, Vol.28 (1), p.113-134
Main Author: Hoefnagel, Michael
Format: Article
Language:English
Subjects:
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Convex and Discrete Geometry
Geometry
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Theorems
Theory of Computation
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Summary: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.
ISSN:0927-2852
1572-9095
DOI:10.1007/s10485-019-09571-z