Triangular Scheme Revisited in the Light of n-permutable Categories

  • Brice Réné Amougou Mbarga University of Yaoundé I, Laboratory of Algebra, Geometry and Application, P.O.Box: 812, Yaoundé, Cameroon
Keywords: categories, n-permutable, triangular scheme


The first diagrammatic scheme was developed by H.P. Gumm under the name Shifting Lemma in case to characterize congruence modularity. A diagrammatic scheme is developed for the generalized semi distributive law in Mal'tsev categories. In this paper we study this diagrammatic scheme in the context of $n$-permutable, and of Mal'tsev categories in particular. Several remarks concerning the Triangular scheme case are included.


M. Barr, Exact categories, in: Exact Categories and Categories of Sheaves, 1-120, Lecture Notes in Math., 236, Springer, Berlin, 1971.

F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and its Applications 566, Kluwer Academic Publishers, Dordrecht, 2004.

S. Burris and H.P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, 78, Springer-Verlag, New York-Berlin, 1981.

A. Carboni, J. Lambek and M.C. Pedicchio, Diagram chasing in Mal’cev categories, Appl. Algebra 69 (1990), 271-284.

A. Carboni, G.M. Kelly and M.C. Pedicchio, Some remarks on Mal’tsev and Goursat categories, Appl. Categ. Structures 1 (1993), 385-421.

I. Chajda and R. Halas, On schemes for congruence distributivity, Open Mathematics 2(3) (2004), 368-376.

I. Chajda, A note on the triangular scheme, East-West J. Math. 3 (2001), 79-80.

I. Chajda and E.K. Horváth, A triangular scheme for congruence distributivity, Acta Sci. Math. (Szeged) 68 (2002), 29-35.

I. Chajda and E.K. Horváth, A scheme for congruence semidistributivity, Discuss. Math. Gen. Algebra Appl. 23 (2003), 13-18.

I. Chajda, E.K. Horváth and G. Czédli, Trapezoid lemma and congruence distributivity, Math. Slovaca 53 (2003), 247-253.

J. Duda, The Triangular Principle for congruence distributive varieties, Abstract of a seminar lecture presented in Brno, March, 2000.

W. Geyer, Generalizing semidistributivity, Order 10 (1903), 77-92.

M. Gran, Notes on regular, exact and additive categories, Summer School on Category Theory and Algebraic Topology, Ecole Polytechnique Fédérale de Lausanne, 11-13 September 2014.

M. Gran and D. Rodelo, A new characterisation of Goursat categories, Appl. Categ. Structures 20 (2012), 229-238. doi:10.1007/s10485-010-9236-x

M. Gran and D. Rodelo, Beck-chavalley condition and Goursat categories, 2013. arxiv:1512.04066v1

M. Gran, D. Rodelo and I. Tchoffo Nguefeu, Variations of the shifting lemma and Goursat categories, Algebra Universalis 80 (2018), Paper No. 2, 12 pp.

M. Gran, D. Rodelo and I. Tchoffo Nguefeu, Facets of congruence distributivity in Goursat categories, 2020. arXiv:1909:10211v2

B. Jonsson Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121.

J. Hagemann and A. Mitschke, On n-permutable congruences, Algebra Universalis 3 (1973), 8-12.

M. Hoefnagel, Majority categories, Theory Appl. Categories 34 (2019), 249-268.

M. Hoefnagel, Characterizations of majority categories, Appl. Categ. Structures 28 (2020), 113-134.

M. Hoefnagel, A categorical approach to lattice-like structures, Ph.D. thesis, 2018.

P.-A. Jacqmin and D. Rodelo, Stability properties characterising n-permutable categories, Theory Appl. Categ. 32 (2017), Paper No. 45, 1563-1587.

K.A. Kearnes and E.W. Kiss, The triangular principle is equivalent to the triangular scheme, Algebra Universalis 54 (2005), 373-383.

H. Peter Gumm, Geometrical methods in congruence modular algebras, Mem. Amer. Math. Soc. 45 (1983).

H. Peter Gumm, The little Desarguesian theorem for algebras in modular varieties, Proc. Amer. Math. Soc. 80(3) (1980), 393-397.

D. Rodelo and T. Van der Linden, Approximation Hagemann-Mitschke co-operations, Appl. Categ. Structures 22 (2014), 1009-1020.

T. Van der Linden and C. Sandry Simeu, Higher extensions in exact Mal’tsev categories: distributivity of congruences and the 3n-lemma, 2018. arXiv:1807:03164v1

How to Cite
Mbarga, B. R. A. (2021). Triangular Scheme Revisited in the Light of n-permutable Categories. Earthline Journal of Mathematical Sciences, 6(1), 105-116.