Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

N-free posets and orthomodularity

Gejza Jenča

Abstract

We prove that the incomparability orthoset of a finite poset is Dacey if and only if the poset is N-free. We give a characterization of finite posets with compatible incomparability orthosets.

N-free posets and orthomodularity

Abstract

We prove that the incomparability orthoset of a finite poset is Dacey if and only if the poset is N-free. We give a characterization of finite posets with compatible incomparability orthosets.
Paper Structure (11 sections, 9 theorems, 16 equations, 2 figures)

This paper contains 11 sections, 9 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Posets, ortholattices
  4. Orthosets
  5. Dacey orthosets
  6. N-free posets
  7. Compatible orthosets
  8. Incomparability orthosets
  9. The orthomodular case
  10. The Boolean case
  11. Conclusion

Key Result

Lemma 2.3

Let $(O,\perp)$ be an orthoset. Let $X,Y$ be subsets of $O$. Then $X$ is orthoclosed and $Y=X^\perp$ if and only if $X\perp Y$ and for every $z\in O\setminus(X\cup Y)$ there are $x\in X$, $y\in Y$ such that $z\not\perp x$ and $z\not\perp y$.

Figures (2)

  • Figure 1: The N
  • Figure 2: The counterexample $P$

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 2.8
  • Definition 3.1
  • ...and 14 more