Congruences of fork extensions of slim semimodular lattices

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 $\boldsymbolγ(S)$ 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 $\boldsymbolγ(S)$ in great detail. The main result follows: \emph{In the order of join-irreducible congruences of a slim, planar, semimodular lattice $L$, the congruence $\boldsymbolγ(S)$ has \emph{at most two covers.}}

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