Planar, infinite, semidistributive lattices

Abstract

An FN lattice $F$ is a simple, infinite, semidistributive lattice. Its existence was recently proved by R. Freese and J.\,B. Nation. Let ${\mathsf{B}}_n$ denote the Boolean lattice with $n$ atoms. For a lattice $K$, let $K^{+}$ denote $K$ with a new unit adjoined. We prove that the finite distributive lattices: ${\mathsf{B}}_0^{+},{\mathsf{B}}_1^{+},{\mathsf{B}}_2^{+},\dots$ can be represented as congruence lattices of infinite semidistributive lattices. The case $n=0$ is the Freese-Nation result, which is utilized in the proof. We also prove some related representation theorems.