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Notes on planar semimodular lattices. VII. Resections of planar semimodular lattices

Gábor Czédli, George Grätzer

Abstract

A recent result of G. Czédli and E.\,T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding "forks". We give a construction that accomplishes the same by deleting elements, by "resections".

Notes on planar semimodular lattices. VII. Resections of planar semimodular lattices

Abstract

A recent result of G. Czédli and E.\,T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding "forks". We give a construction that accomplishes the same by deleting elements, by "resections".

Paper Structure

This paper contains 5 sections, 11 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The construction
  3. The results
  4. Schemes
  5. The proof of the main result

Key Result

Theorem 1

Slim semimodular lattice diagrams are characterized as diagrams obtained from slim distributive lattice diagrams by a sequence of resections.

Figures (5)

  • Figure 1: Resect this diagram at the element marked by the big circle by deleting the black-filled elements
  • Figure 3: $\mathsf{N}_{7}$ and its variants
  • Figure 4: Some slim semimodular diagrams
  • Figure 5: The process does not stop
  • Figure 6: Insertion at $u$ ($t_9$ plays a role only in Case 3)

Theorems & Definitions (12)

  • Theorem 1
  • Remark 2
  • Corollary 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Corollary 9
  • Lemma 10
  • ...and 2 more