Abstract
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 \(A_1 \times A_2 \times \cdots \times A_n\) 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 \(\alpha \cap (\beta \circ \gamma ) = (\alpha \cap \beta ) \circ (\alpha \cap \gamma )\) due to A.F. Pixley.
Similar content being viewed by others
References
Baker, K.A., Pixley, A.F.: Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143, 165–174 (1975)
Barr, M., Grillet, P.A., van Osdol, D.H.: Exact Categories and Categories of Sheaves. Lecture Notes in Mathematics, vol. 236. Springer, Berlin (1971)
Birkhoff, G., Kiss, S.A.: A ternary operation in distributive lattices. Bull. Am. Math. Soc. 53(8), 749–752 (1947)
Bourn, D.: A categorical genealogy for the congruence distributive property. Theory Appl. Categ. 8, 391–407 (2001)
Bourn, D.: Intrinsic centrality and associated classifying properties. J. Algebra 256, 126–145 (2002)
Bourn, D.: Congruence distributivity in Goursat and Mal’cev categories. Appl. Categ. Struct. 13, 101–111 (2005)
Bourn, D., Janelidze, Z.: Approximate Mal’tsev operations. Theory Appl. Categ. 21, 152–171 (2008)
Carboni, A., Lambek, J., Pedicchio, M.C.: Diagram chasing in Mal’cev categories. J. Pure Appl. Algebra 69, 271–284 (1991)
Carboni, A., Pedicchio, M.C., Pirovano, N.: Internal graphs and internal groupoids in Mal’cev categories. In: CMS Conference Proceedings, vol. 13, pp. 97–109 (1991)
Hoefnagel, M.A.: A categorical approach to lattice-like structures. PhD Thesis, University of Stellenbosch (2018)
Hoefnagel, M.A.: Majority categories. Theory Appl. Categ. 34(10), 249–268 (2019)
Janelidze, G.: A history of selected topics in categorical algebra I: from Galois theory to abstract commutators and internal groupoids. Categ. Gen. Algebr. Struct. Appl. 8, 1–54 (2016)
Janelidze, Z.: Generalized difunctionality, Pixley categories, and a general Bourn localization theorem. In: 67th Workshop on General Algebra (2004)
Mitschke, A.: Near unanimity identities and congruence distributivity in equational classes. Algebra Universalis 8, 29–38 (1978)
Pixley, A.F.: Distributivity and permutability of congruence relations in equational classes of algebras. Proc. Am. Math. Soc. 14, 105–109 (1963)
Wille, R.: Kongruenzklassengeometrien. Lecture Notes in Mathematics. Springer, Berlin (1970)
Acknowledgements
Many thanks are due to Prof. Z. Janelidze, for many helpful and stimulating discussions on the topic of regular majority categories.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dominique Bourn.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hoefnagel, M. Characterizations of Majority Categories. Appl Categor Struct 28, 113–134 (2020). https://doi.org/10.1007/s10485-019-09571-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-019-09571-z
Keywords
- Majority category
- Arithmetical category
- Congruence distributivity
- Characterizations of majority categories