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Meet-irreducible congruence lattices

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Abstract

The system of all congruences of an algebra (A, F) forms a lattice, denoted \({{\mathrm{Con}}}(A, F)\). Further, the system of all congruence lattices of all algebras with the base set A forms a lattice \(\mathcal {E}_A\). We deal with meet-irreducibility in \(\mathcal {E}_A\) for a given finite set A. All meet-irreducible elements of \(\mathcal {E}_A\) are congruence lattices of monounary algebras. Some types of meet-irreducible congruence lattices were described in Jakubíková-Studenovská et al. (2017). In this paper, we prove necessary and sufficient conditions under which \({{\mathrm{Con}}}(A, f)\) is meet-irreducible in the case when (A, f) is an algebra with short tails (i.e., f(x) is cyclic for each \(x \in A\)) and in the case when (A, f) is an algebra with small cycles (every cycle contains at most two elements).

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Authors and Affiliations

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

    Danica Jakubíková–Studenovská & Lucia Janičková

Authors
  1. Danica Jakubíková–Studenovská
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  2. Lucia Janičková
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Correspondence to Danica Jakubíková–Studenovská.

Additional information

Presented by R. Pöschel.

This work was supported by Grant VEGA 1/0097/18.

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Jakubíková–Studenovská, D., Janičková, L. Meet-irreducible congruence lattices. Algebra Univers. 79, 89 (2018). https://doi.org/10.1007/s00012-018-0571-3

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  • DOI: https://doi.org/10.1007/s00012-018-0571-3

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