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Algebra universalis

, 79:9 | Cite as

Characterizing fully principal congruence representable distributive lattices

  • Gábor Czédli
Article
Part of the following topical collections:
  1. In memory of Bjarni Jónsson

Abstract

Motivated by a recent paper of G. Grätzer, a finite distributive lattice D is called fully principal congruence representable if for every subset Q of D containing 0, 1, and the set J(D) of nonzero join-irreducible elements of D, there exists a finite lattice L and an isomorphism from the congruence lattice of L onto D such that Q corresponds to the set of principal congruences of L under this isomorphism. A separate paper of the present author contains a necessary condition of full principal congruence representability: D should be planar with at most one join-reducible coatom. Here we prove that this condition is sufficient. Furthermore, even the automorphism group of L can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Grätzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.

Keywords

Distributive lattice Principal lattice congruence Congruence lattice Principal congruence representable Simultaneous representation Automorphism group 

Mathematics Subject Classification

06B10 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary

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