Congruences of fork extensions of slim semimodular lattices

2013/07/31 by George Grätzer, Grätzer, George
Mathematics · #06B10 #06C10 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:06B10 #msc:06C10

paper · pdf · doi:10.48550/arxiv.1307.8404

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

arxiv created 2014/03/03 · arxiv updated 2014/03/04

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 \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: \emphIn the order of join-irreducible congruences of a slim, planar, semimodular lattice L, the congruence \boldsymbolγ(S) has at most two covers.

Citations

• Congruences of the fork extensions. II. The congruence gamma
• Congruences of the fork extensions. I. The Congruence Extension Property

Cited by

• Congruences of the fork extensions. I. The Congruence Extension Property
• Congruences of the fork extensions. II. The congruence gamma

Related

• Congruences of the fork extensions. I. The Congruence Extension Property
• Congruences of the fork extensions. II. The congruence gamma
• An open problem on congruences of finite lattices
• A new property of congruence lattices of slim, planar, semimodular lattices
• Using the Swing Lemma and Czédli diagrams for congruences of planar semimodular lattices