Congruences of fork extensions of slim semimodular lattices

TL;DR

This paper studies how congruences extend in fork extensions of slim, planar, semimodular lattices, introducing a new join-irreducible congruence and analyzing its properties and cover relations.

Contribution

It introduces a new join-irreducible congruence in fork extensions and characterizes when it is new and how it relates to existing congruences.

Findings

The congruence $oldsymbol{ angle}$ has at most two covers in the congruence lattice.

Conditions are identified for $oldsymbol{ angle}$ to be new.

Detailed description of the new congruence $oldsymbol{ angle}$ when it is not generated by existing congruences.

Abstract

For a slim, planar, semimodular lattice $L$ and covering square~$S$, G.~Cz\'edli and E.\,T.~Schmidt introduced the fork extension, $L[S]$, which is also a slim, planar, semimodular lattice. We investigate when a congruence of $L$ extends to $L[S]$. We introduce a join-irreducible congruence $\boldsymbol{\gamma}(S)$ of $L[S]$. We determine when it is new, in the sense that it is not generated by a join-irreducible congruence of $L$. When it is new, we describe the congruence $\boldsymbol{\gamma}(S)$ in great detail. The main result follows: \emph{In the order of join-irreducible congruences of a slim, planar, semimodular lattice $L$, the congruence $\boldsymbol{\gamma}(S)$ has \emph{at most two covers.}}