# 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.

## Keywords

Distributive lattice Principal lattice congruence Congruence lattice Principal congruence representable Simultaneous representation Automorphism group## Mathematics Subject Classification

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