Lamps in slim rectangular planar semimodular lattices

Czédli, Gábor

doi:10.14232/actasm-021-865-y

Cited by 11 publications

(8 citation statements)

References 21 publications

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“…G. Czédli [13] discovered four major properties of the congruence lattices of SPS lattices. We'll discuss them in Section 8.…”

Section: The Two-cover Propertymentioning

confidence: 99%

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On a Property of Congruence Lattices of Slim, Planar, Semimodular Lattices

Grätzer

2023

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In a 2121 paper with G´abor Cz´edli, we introduced and verified the Three-pendant Three-crown Property, 3P3C, for congruence lattices of slim (no M3 sublattice), planar, semimodular lattices. The proof is very long; in part, because it relies on Cz´edli’s 2021 paper on lamps. This paper verifies 3P3C using the Swing Lemma, an elementary and short approach. 2020 Mathematics Subject Classification. 06C10.

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“…G. Czédli [13] discovered four major properties of the congruence lattices of SPS lattices. We'll discuss them in Section 8.…”

Section: The Two-cover Propertymentioning

confidence: 99%

“…On the edges of L with meet-irreducible bottoms, the internal swing, p in q, is a reflexive, symmetric, and transitive binary relation. So these edges are partitioned into equivalence classes; G. Czédli [13] calls them lamps.…”

Section: The Two-cover Propertymentioning

confidence: 99%

On a Property of Congruence Lattices of Slim, Planar, Semimodular Lattices

Grätzer

2023

Preprint

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“…A number of papers developed tools to tackle this problem: the Swing Lemma (G. Grätzer [14]), trajectory coloring (G. Czédli [1]), special diagrams (G. Czédli [3]), lamps (G. Czédli [4]). Some of these results require long proofs.…”

Section: Motivationmentioning

confidence: 99%

“…G. Czédli [4,Corollaries 3.4,3.5,Theorem 4.3] found four new properties of congruence lattices of slim, planar, semimodular lattices.…”

Section: Introductionmentioning

confidence: 99%

Using the Swing Lemma and $\mathcal{C}_1$-diagrams for congruences of planar semimodular lattices

Grätzer

2021

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A planar semimodular lattice K is slim if M 3 is not a sublattice of K. In a recent paper, G. Czédli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the No Child Property: Let P be the ordered set of join-irreducible congruences of K. Let x, y, z ∈ P and let z be a maximal element of P. If x = y and x, y ≺ z in P, then there is no element u of P such that u ≺ x, y in P.We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Czédli's, to verify his four properties.

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“…C 1 -diagrams. This research tool, introduced by G. Czédli, has been playing an important role in some recent papers, see G. Czédli [1], [2], and [3], G. Czédli and G. Grätzer [4], and G. Grätzer [7]; for the definition, see G. Czédli [1] and G. Grätzer [7].…”

Section: Introductionmentioning

confidence: 99%

Another research note

Grätzer¹

2022

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A planar semimodular lattice L is slim if M 3 is not a sublattice of L. In a recent paper, G. Czédli introduced a very powerful diagram type for slim, planar, semimodular lattices, the C 1 -diagrams. Part IX provides a very short proof of the existence of C 1 -diagrams.In this part, we revisit the concept of natural diagrams I introduced with E. Knapp about a dozen years ago. Given a slim rectangular lattice L, we construct its natural diagram in one simple step. In this note, we show that for a slim rectangular lattice, a natural diagram is the same as a C 1 -diagram. This verifies that natural diagrams have all the nice properties of C 1 -diagrams.

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Lamps in slim rectangular planar semimodular lattices

Cited by 11 publications

References 21 publications

On a Property of Congruence Lattices of Slim, Planar, Semimodular Lattices

On a Property of Congruence Lattices of Slim, Planar, Semimodular Lattices

Using the Swing Lemma and $\mathcal{C}_1$-diagrams for congruences of planar semimodular lattices

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