Some Applications of the Swing Lemma

George Grätzer

Chapter Chapter 25 in The Congruences of a Finite Lattice, 2016, pp 315-321 from Springer

Abstract: Abstract We start with an important definition of Czédli, G. G. Czédli [18]. For the trajectories 𝒫 ≠ 𝒬 $$\mathcal{P}\neq \mathcal{Q}$$ , let 𝒫 ≤ C 𝒬 $$\mathcal{P}\leq _{C}\mathcal{Q}$$ if 𝒫 $$\mathcal{P}$$ is a hat trajectory, 1 top ( 𝒫 ) ≤ 1 top ( 𝒬 ) $$1_{\text{top}(\mathcal{P})} \leq 1_{\text{top}(\mathcal{Q})}$$ , and 0 top ( 𝒫 ) ≰ 0 top ( 𝒬 ) $$0_{\text{top}(\mathcal{P})}\nleq0_{\text{top}(\mathcal{Q})}$$ , see Figure 25.1. Czédli defines ≤ T as the reflexive and transitive Reflexive-transitive closure closure of ≤ C . (The notation in G. Czédli [18] is different.) So for a trajectory 𝒫 $$\mathcal{P}$$ , we can define the closure, 𝒫 ̂ $$\widehat{\mathcal{P}}$$ , of 𝒫 $$\mathcal{P}$$ : 𝒬 ∈ 𝒫 ̂ $$\mathcal{Q}\in \widehat{\mathcal{P}}$$ iff 𝒫 ≤ C 𝒬 $$\mathcal{P}\leq _{C}\mathcal{Q}$$ and 𝒬 ≤ C 𝒫 $$\mathcal{Q}\leq _{C}\mathcal{P}$$ .

Keywords: Join-irreducible Congruences; Trajectory Theorem; Planar Semimodular Lattices; Prime Interval; Congruence-preserving Extension (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-38798-7_25

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