The largest higher commutator sequence

Nebojša Mudrinski


Given the congruence lattice L of a finite algebra A that generates a congruence permutable variety, we  look for those sequences of operations on L that have the properties of higher commutator operations of expansions of A. If we introduce the order of such sequences in the natural way the question is whether exists or not the largest one. The answer is positive. We provide a description of the largest element and as a consequence we obtain that the sequences form a complete lattice.

Received 18 September 2018

Supported by the Austrian Science Fund (FWF):P29931 and the Scientific Project 174018 of the Ministry of Science and Education of the Republic of Serbia.

AMS subject classification: Primary 06B10; Secondary 06A07, 08A40

Słowa kluczowe: lattices, sequences of operations, commutators

[1] E. Aichinger and N. Mudrinski, Some applications of higher commutators in Mal’cev algebras, Algebra Universalis 63:4 (2010), 367–403.

[2] E. Aichinger and N. Mudrinski, Sequences of commutator operations, Order  30 (2013), 859–867.

[3] S. Burris and H.P. Sankappanavar, A course in universal algebra, Springer, New York, Heidelberg, Berlin, 1981.

[4] G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, Vol. 25, rev. edn., Americal Mathematical Society, New York, 1948.

[5] A. Bulatov, On the number of finite Mal’tsev algebras, Contributions to general algebra, 13 (Velk´e Karlovice, 1999 / Dresden, 2000), Heyn, Klagenfurt, 2001, pp. 41–54.

[6] J. Czelakowski, Additivity of the commutator and residuation, Rep. Math. Logic 43 (2008), 109–132.

[7] J. Czelakowski, The Equationally-Defined Commutator: A study in equational logic and algebra, Birkhäuser /Springer, Cham, 2015.

[8] J. Czelakowski, The equationally defined commutator in quasivarieties generated by two-element algebras, Outstanding Contributions to Logic 16, Don Pigozzi on Abstract Algebraic Logic, Universal Algebra and Computer Science, Springer, Cham, 2018, pp. 131–165.

[9] R. Freese and R.N. McKenzie, Commutator theory for congruence modular varieties, London Math. Soc. Lecture Note Ser., Vol. 125, Cambridge University Press, 1987.

[10] R.N. McKenzie, G.F. McNulty, and W.F. Taylor, Algebras, lattices, varieties, Vol. 1, Wadsworth & Brooks / Cole Advanced Books & Software, Monterey, California, 1987.

[11] A. Moorhead, Higher commutator theory for congruence modular varieties, Journal of Algebra 513 (2018), 133–158.

[12] N. Mudrinski, On Polynomials in Mal’cev Algebras, Ph.D. thesis, University of Novi Sad, 2009. Available at:˜nmudrinski/DissertationMudrinski.pdf.

[13] J.D.H. Smith, Mal’cev varieties, Lecture Notes in Math., Vol. 554, Springer Verlag, Berlin, 1976.

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.