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arXiv · 2306.04113

Planar, infinite, semidistributive lattices

George Grätzer·J. B. Nation
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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.

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George Grätzer, J. B. Nation. 2023-06-07. Planar, infinite, semidistributive lattices. https://arxiv.org/abs/2306.04113

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