Algebra universalis

, 79:4 | Cite as

The lattice of congruence lattices of algebras on a finite set

  • Danica Jakubíková-Studenovská
  • Reinhard Pöschel
  • Sándor Radeleczki
Article
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Part of the following topical collections:
  1. In memory of E. Tamás Schmidt

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 \(|A|\ge 4\).

Keywords

Congruence lattice Unary operation Monounary algebra Join-irreducible element Meet-irreducible element Tolerance simple 

Mathematics Subject Classification

Primary 08A30 Secondary 06B15 08A60 06A15 08A35 08A99 20M20 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Danica Jakubíková-Studenovská
    • 1
  • Reinhard Pöschel
    • 2
  • Sándor Radeleczki
    • 3
  1. 1.Institute of MathematicsP. J. Šafárik UniversityKošiceSlovakia
  2. 2.Institute of AlgebraTU DresdenDresdenGermany
  3. 3.Institute of MathematicsUniversity of MiskolcMiskolcHungary

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