Abstract
We show that varietal techniques based on the existence of operations of a certain arity can be extended to n-permutable categories with binary coproducts. This is achieved via what we call approximate Hagemann–Mitschke co-operations, a generalisation of the notion of approximate Mal’tsev co-operation [2]. In particular, we extend characterisation theorems for n-permutable varieties due to J. Hagemann and A. Mitschke [8, 9] to regular categories with binary coproducts.
References
Barr, M.: Exact categories. Exact categories and categories of sheaves. Lecture Notes in Mathematics, vol. 236, pp. 1–120. Springer (1971)
Bourn, D., Janelidze, Z.: Approximate Mal’tsev operations. Theory Appl. Categ. 21(8), 152–171 (2008)
Bourn, D., Janelidze, Z.: Pointed protomodularity via natural imaginary subtractions. J. Pure Appl. Algebra 213, 1835–1851 (2009)
Bourn, D., Janelidze, Z.: Subtractive categories and extended subtractions. Appl. Categ. Structures 17, 317–343 (2009)
Bourn, D., Janelidze, Z.: Categorical (binary) difference terms and protomodularity. Algebra Univers. 66, 277–316 (2011)
Carboni, A., Kelly, G.M., Pedicchio, M.C.: Some remarks on Maltsev and Goursat categories. Appl. Categ. Struct. 1, 385–421 (1993)
Grätzer, G.: Two Mal’cev type theorems in universal algebra. J. Comb. Theory 8, 334–342 (1970)
Hagemann, J.: Grundlagen der allgemeinen topologischen Algebra, unpublished
Hagemann, J., Mitschke, A.: On n-permutable congruences. Algebra Univers. 3, 8–12 (1973)
Janelidze, Z.: Closedness properties of internal relations VI: approximate operations. Cah. Topol. Géom. Différ. Catégor. L, 298–319 (2009)
Rodelo, D., Janelidze, Z., Van der Linden, T.: Hagemann’s theorem for regular categories, J. Homotopy Relat. Struct. (2013) in press
Mal’cev, A.I.: On the general theory of algebraic systems. Mat. Sbornik N. S. 35(6), 3–20 (1954)
Martins-Ferreira, N., Van der Linden, T.: Categories vs. Groupoids via Generalised Mal’tsev Properties arXiv:1206.2745v1 (2012)
Mitschke, A.: Implication algebras are 3-permutable and 3-distributive. Algebra Univers. 1, 182–186 (1971)
Schmidt, E.T.: On n-permutable equational classes. Acta Sci. Math. (Szeged) 33, 29–39 (1972)
Smith, J.D.H.: Mal’cev varieties. Lecture Notes in Mathematics, vol. 554. Springer (1976)
Wille, R.: Kongruenzklassengeometrien. Lecture Notes in Mathematics, vol. 113. Springer (1970)
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Dedicated to George Janelidze on the occasion of his sixtieth birthday
The first author was supported by Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT–Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2013 and grants number PTDC/MAT/120222/2010 and SFRH/BPD/69661/2010. The second author works as chercheur qualifié for Fonds de la Recherche Scientifique–FNRS and would like to thank CMUC for its kind hospitality during his stays in Coimbra.
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Rodelo, D., Van der Linden, T. Approximate Hagemann–Mitschke Co-operations. Appl Categor Struct 22, 1009–1020 (2014). https://doi.org/10.1007/s10485-013-9359-y
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DOI: https://doi.org/10.1007/s10485-013-9359-y