Characterizing fully principal congruence representable distributive\n lattices

2017/06/11 by Gábor Czédli, Czédli, Gábor
Computer Science · #Advanced Algebra and Logic

paper · pdf · doi:10.48550/arxiv.1706.03401

Abstract

Motivated by a recent paper of G. Gr "atzer, a finite distributive lattice\nD is said to be fully principal congruence representable if for every subset\nQ of D containing 0, 1, and the set J(D) of nonzero join-irreducible\nelements of D, there exists a finite lattice L and an isomorphism from the\ncongruence lattice of L onto D such that Q corresponds to the set of\nprincipal congruences of L under this isomorphism. Based on earlier results\nof G. Gr "atzer, H. Lakser, and the present author, we prove that a finite\ndistributive lattice D is fully principal congruence representable if and\nonly if it is planar and it has at most one join-reducible coatom. Furthermore,\neven the automorphism group of L can arbitrarily be stipulated in this case.\nAlso, we generalize a recent result of G. Gr "atzer on principal congruence\nrepresentable subsets of a distributive lattice whose top element is\njoin-irreducible by proving that the automorphism group of the lattice we\nconstruct can be arbitrary.\n

Related

• Characterizing representability by principal congruences for finite\n distributive lattices with a join-irreducible unit element
• On the set of principal congruences in a distributive congruence lattice of an algebra
• Revisiting the representation theorem of finite distributive lattices with principal congruences
• Some preliminary results on the set of principal congruences of a finite lattice
• Minimal representations of a finite distributive lattice by principal\n congruences of a lattice