Congruences of fork extensions of slim, planar, semimodular lattices

Abstract

For a slim, planar, semimodular lattice L and a covering square S of L, G. Czédli and E. T. Schmidt introduced the fork extension, L[S], which is also a slim, planar, semimodular lattice. This paper investigates when a congruence of L extends to L[S]. We introduce a join-irreducible congruence $${\gamma}$$ (S) of L[S] and determine when it is new, in the sense that it is not generated by a join-irreducible congruence of L. We provide a complete description of $${\gamma}$$ (S). Then we prove that $${\gamma}$$ (S) has at most two covers in the order of join-irreducible congruences of L[S]. Finally, we derive the main result of this paper, the Two-cover Theorem: Every join-irreducible congruence has at most two covers in the order of join-irreducible congruences of a slim, planar, semimodular lattice L.