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Algebra universalis

, 79:89 | Cite as

Meet-irreducible congruence lattices

  • Danica Jakubíková–Studenovská
  • Lucia Janičková
Article
  • 17 Downloads

Abstract

The system of all congruences of an algebra (AF) 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 (Af) is an algebra with short tails (i.e., f(x) is cyclic for each \(x \in A\)) and in the case when (Af) is an algebra with small cycles (every cycle contains at most two elements).

Mathematics Subject Classification

08A30 08A60 08A62 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Danica Jakubíková–Studenovská
    • 1
  • Lucia Janičková
    • 1
  1. 1.Institute of MathematicsP.J. Šafárik University in KošiceKošiceSlovakia

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