Characterizing fully principal congruence representable distributive lattices
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 joinirreducible elements of D, there exists a finite lattice L and an isomorphism from the congruence...
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Format:  Article 
Published: 
2018

Series:  ALGEBRA UNIVERSALIS
79 No. 1 
doi:  10.1007/s0001201804988 
mtmt:  3362775 
Online Access:  http://publicatio.bibl.uszeged.hu/14522 
Summary:  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 joinirreducible 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 joinreducible 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 joinirreducible by proving that the automorphism group of the lattice we construct can be arbitrary. © 2018, Springer International Publishing AG, part of Springer Nature. 

ISSN:  00025240 