DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in volume

Authors:

G. Grätzer

George Grätzer

Department of Mathematics
University of Manitoba
Winnipeg, MB R3T 2N2, Canada

email: gratzer@me.com

Title:

Using the Swing Lemma and $\boldsymbol{\mathcal{C}_1}$-diagrams for congruences of planar semimodular lattices

PDF

Source:

Discussiones Mathematicae - General Algebra and Applications 43(1) (2023) 63-74

Received: 2021-06-06 , Revised: 2021-07-18 , Accepted: 2021-07-19 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1410

Abstract:

A planar semimodular lattice $K$ is slim if $\mathsf{M}_{3}$ is not a sublattice of $K$. In a recent paper, G. Czédli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the No Child Property: Let $\mathcal{P}$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in \mathcal{P}$ and let $z$ be a maximal element of $\mathcal{P}$. If $x \neq y$ and $x, y \prec z$ in $\mathcal{P}$, then there is no element $u$ of $\mathcal{P}$ such that $u \prec x, y$ in $\mathcal{P}$. The Swing Lemma and a standardized diagram type are used to give direct proofs of Czédli's four properties.

Keywords:

rectangular lattice, slim planar semimodular lattice, congruence lattice

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