Congruences of fork extensions of lattices
George Grätzer · arXiv (Cornell University) · 2013
For a slim, planar, semimodular lattice $L$ and covering square $S$, G. Cz\'edli and E.\,T. Schmidt introduced the fork extension $L[S]$. We are going to investigate the congruences of $L[S]$. We prove that a fork extension has the Congruence Extension Property: every congruence of the lattice $L$ extends to $L[S]$. We prove that either $L[S]$ is a congruence-preserving extension of $L$ or it has exactly one new join-irreducible congruence $\bgg(S)$. We are going to examine in detail this congruence.