Congruences of fork extensions of slim semimodular lattices

Abstract

For a slim, planar, semimodular lattice $L$ and covering square~$S$, G.~Czédli 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 $\mathbf{\gamma }(S)$ 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 $\mathbf{\gamma }(S)$ in great detail. The main result follows: \emph{In the order of join-irreducible congruences of a slim, planar, semimodular lattice $L$, the congruence $\mathbf{\gamma }(S)$ has \emph{at most two covers.}}

arXiv commentarXiv admin note: substantial text overlap with arXiv:1307.6126, arXiv:1307.0778