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Notes on a short-cut to the proof of the $\mathbf{M}_3$-$\mathbf{N}_5$ Theorem

M. R. Emamy-K., Gustavo A. Meléndez Ríos

Abstract

This paper presents two shortcuts to a classical proof of the $\mathbf{M}_3$-$\mathbf{N}_5$ Theorem, which can be found in B. Davey and H. Priestley [2] and S. Burris and H. Sankappanavar [1]. To be precise, the shortcuts pertain a particular step of the proof that requires showing an algebraic equality. In addition, we briefly discuss how to compare the lengths of the three proofs (the original and our two proposed shortcuts). To do so, we introduce two methods to compare the lengths of proofs based on algebraic lattice expressions. We call them the proof count method and the proof poset method. Both methods indicate that our proofs are shorter but the difference is more pronounced in the former. Keywords: lattices, posets

Notes on a short-cut to the proof of the $\mathbf{M}_3$-$\mathbf{N}_5$ Theorem

Abstract

This paper presents two shortcuts to a classical proof of the - Theorem, which can be found in B. Davey and H. Priestley [2] and S. Burris and H. Sankappanavar [1]. To be precise, the shortcuts pertain a particular step of the proof that requires showing an algebraic equality. In addition, we briefly discuss how to compare the lengths of the three proofs (the original and our two proposed shortcuts). To do so, we introduce two methods to compare the lengths of proofs based on algebraic lattice expressions. We call them the proof count method and the proof poset method. Both methods indicate that our proofs are shorter but the difference is more pronounced in the former. Keywords: lattices, posets
Paper Structure (9 sections, 3 theorems, 6 equations, 2 figures, 3 tables)

This paper contains 9 sections, 3 theorems, 6 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. $\mathbf{M}_3$-$\mathbf{N}_5$ Theorem
  5. 3 Proofs of $u \land v = p$
  6. Comparing Proof Lengths
  7. Proof Count Method
  8. Proof Poset Method
  9. Conclusion

Key Result

Lemma 1

(Connecting Lemma) Let $L$ be a lattice with $a,b \in L$ and induced order $\leq$. Then the following are equivalent:

Figures (2)

  • Figure 1: The diamond $\mathbf{M}_3$
  • Figure 2: The pentagon $\mathbf{N}_5$

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Theorem 3
  • Proof 1
  • Proof 2
  • Proof 3
  • Definition 4
  • Definition 5