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Congruences of fork extensions of slim, planar, semimodular lattices

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    G. Grätzer

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  1. G. Grätzer
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Correspondence to G. Grätzer.

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Presented by B. Davey.

To the memory of my friends, Ervin Fried and Jiří Sichler

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Grätzer, G. Congruences of fork extensions of slim, planar, semimodular lattices. Algebra Univers. 76, 139–154 (2016). https://doi.org/10.1007/s00012-016-0394-z

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