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The lattice of quasiorder lattices of algebras on a finite set

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Abstract

The quasiorders of an algebra (A, F) constitute a common generalization of its congruences and compatible partial orders. The quasiorder lattices of all algebras defined on a fixed set A ordered by inclusion form a complete lattice \({\mathcal{L}}\). The paper is devoted to the study of this lattice \({\mathcal{L}}\). We describe its join-irreducible elements and its coatoms. Each meet-irreducible element of \({\mathcal{L}}\) 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 characterizations, we deduce several properties of the lattice \({\mathcal{L}}\); in particular, we prove that \({\mathcal{L}}\) is always tolerance-simple.

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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 J. Berman.

The research of the first author was partially supported by Slovak VEGA grant 1/0063/14. The second author was partially supported by DFG grant 436 UNG. The research of the third author 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 quasiorder lattices of algebras on a finite set. Algebra Univers. 75, 197–220 (2016). https://doi.org/10.1007/s00012-016-0373-4

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  • DOI: https://doi.org/10.1007/s00012-016-0373-4

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