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Characterizing fully principal congruence representable distributive lattices

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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.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Gábor Czédli

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  1. Gábor Czédli
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Correspondence to Gábor Czédli.

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Dedicated to the memory of Bjarni Jónsson.

Presented by J.B. Nation.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.

This research was supported by NFSR of Hungary (OTKA), Grant number K 115518.

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Czédli, G. Characterizing fully principal congruence representable distributive lattices. Algebra Univers. 79, 9 (2018). https://doi.org/10.1007/s00012-018-0498-8

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