English

Planar, infinite, semidistributive lattices

Rings and Algebras 2023-09-26 v3

Abstract

An FN lattice FF is a simple, infinite, semidistributive lattice. Its existence was recently proved by R. Freese and J.\,B. Nation. Let Bn\mathsf{B}_n denote the Boolean lattice with nn atoms. For a lattice KK, let K+K^+ denote KK with a new unit adjoined. We prove that the finite distributive lattices: B0+,B1+,B2+,\mathsf{B}_0^+, \mathsf{B}_1^+,\mathsf{B}_2^+, \dots can be represented as congruence lattices of infinite semidistributive lattices. The case n=0n = 0 is the Freese-Nation result, which is utilized in the proof. We also prove some related representation theorems.

Keywords

Cite

@article{arxiv.2306.04113,
  title  = {Planar, infinite, semidistributive lattices},
  author = {George Grätzer and J. B. Nation},
  journal= {arXiv preprint arXiv:2306.04113},
  year   = {2023}
}
R2 v1 2026-06-28T10:58:23.850Z