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The lattice of congruence lattices of algebras on a finite set

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Abstract

The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice \(\mathcal {E}\). We describe the atoms and coatoms. Each meet-irreducible element of \(\mathcal {E}\) being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice \(\mathcal {E}\); in particular, we prove that \(\mathcal {E}\) is tolerance-simple whenever \(|A|\ge 4\).

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Authors and Affiliations

  1. Institute of Mathematics, P. J. Šafárik University, Košice, Slovakia

    Danica Jakubíková-Studenovská

  2. Institute of Algebra, TU Dresden, Dresden, Germany

    Reinhard Pöschel

  3. Institute of Mathematics, University of Miskolc, Miskolc, Hungary

    Sándor Radeleczki

Authors
  1. Danica Jakubíková-Studenovská
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  2. Reinhard Pöschel
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  3. Sándor Radeleczki
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Corresponding author

Correspondence to Reinhard Pöschel.

Additional information

Presented by G. Czedli.

Dedicated to E. Tamás Schmidt.

This article is part of the topical collection “In memory of E. T. Schmidt” edited by Robert W. Quackenbush.

The research of Danica Jakubíková-Studenovská was partially supported by Slovak VEGA Grant 1/0063/14. The research of Sándor Radeleczki started as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project, supported by the European Union, co-financed by the European Social Fund 113/173/0-2.

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Jakubíková-Studenovská, D., Pöschel, R. & Radeleczki, S. The lattice of congruence lattices of algebras on a finite set. Algebra Univers. 79, 4 (2018). https://doi.org/10.1007/s00012-018-0486-z

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