Complete congruence lattices of two related modular lattices

Abstract

By a 1991 result of R. Freese, G. Grätzer, and E. T. Schmidt, every complete lattice A is isomorphic to the lattice Com(K) of complete congruences of a strongly atomic, 3-distributive, complete modular lattice K. In 2002, Grätzer and Schmidt improved 3-distributivity to 2-distributivity. Here, we represent morphisms between two complete lattices with complete lattice congruences in three ways. Namely, for ${i \in \{1, 2, 3\},}$ let ${A_i}$ and ${A^{\prime}_{i}}$ be arbitrary complete lattices and let ${f_i}$ : ${A_{i} \rightarrow A^{\prime}_{i}}$ be maps such that (i) ${f_1}$ is ${(\bigvee, 0)}$-preserving and 0-separating, (ii) ${f_2}$ is ${(\bigwedge, 0, 1)}$-preserving, and (iii) ${f_3}$ is ${(\bigvee, 0)}$-preserving. We prove that for ${i \in \{1, 2, 3\},}$ there exist strongly atomic, 2-distributive, complete modular lattices ${K_i}$ and ${K^{\prime}_{i}}$ such that ${A_{i} \cong {\rm Com}({K_{i}}), A^{\prime}_{i} \cong {\rm Com}({K^{\prime}_{i}}),}$ and, in addition, (i) ${K_1}$ is a principal ideal of ${{K^{\prime}_{1}}}$ and ${f_1}$ is represented by complete congruence extension, (ii) ${K^{\prime}_{2}}$ is a sublattice of ${K_2}$ and ${f_2}$ is represented by restriction, and (iii) ${f_3}$ is represented as the composite of a map naturally induced by a complete lattice homomorphism from ${K_3}$ to ${K^{\prime}_{3}}$ and the complete congruence generation in ${K^{\prime}_{3}}$. Also, our approach yields a relatively easy construction that proves the above-mentioned 2002 result of Grätzer and Schmidt.