Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantales carrying ortholattice structure

Michal Botur, David Kruml, Jan Paseka

Abstract

This paper investigates the intersection of residuated structures from many-valued logic and orthomodular lattices from quantum logic. We explore whether non-Boolean structures can simultaneously satisfy residuation principles and orthocomplementation requirements. Our main contribution is a study of Girard posets with inversions, providing a characterization theorem where a unital residuated poset is Girard if and only if it admits an inversion satisfying specific adjointness conditions. We prove that any complemented lattice admitting an integral residuated structure must be Boolean, which motivates our search for orthomodular examples in the non-integral case. We answer this by demonstrating that the lattice $C(\mathbb{R}^n)$ of closed subspaces of $n$-dimensional real coordinate space carries both an orthomodular and a commutative Girard quantale structure. This construction provides a concrete non-Boolean framework unifying quantum-logical and many-valued logical reasoning.

Quantales carrying ortholattice structure

Abstract

This paper investigates the intersection of residuated structures from many-valued logic and orthomodular lattices from quantum logic. We explore whether non-Boolean structures can simultaneously satisfy residuation principles and orthocomplementation requirements. Our main contribution is a study of Girard posets with inversions, providing a characterization theorem where a unital residuated poset is Girard if and only if it admits an inversion satisfying specific adjointness conditions. We prove that any complemented lattice admitting an integral residuated structure must be Boolean, which motivates our search for orthomodular examples in the non-integral case. We answer this by demonstrating that the lattice of closed subspaces of -dimensional real coordinate space carries both an orthomodular and a commutative Girard quantale structure. This construction provides a concrete non-Boolean framework unifying quantum-logical and many-valued logical reasoning.
Paper Structure (8 sections, 9 theorems, 32 equations)

This paper contains 8 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Inversions, ortholattices and orthomodular lattices
  4. Residuated posets
  5. Quantales
  6. Girard posets and inversions
  7. Quantales carrying ortholattice structure
  8. Conclusions

Key Result

Lemma 4

Jac Let $X$ be an orthomodular lattice, with element $a\in X$. The (principal) downset $\mathop{\downarrow}\! a = \{u\in X\;|\;u \leq a\}$ is again an orthomodular lattice, with order, meets and joins as in $X$, but with its own orthocomplement $\perp_a$ given by $u^{\perp_a} = a \mathrel{\wedge} u^

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Example 3
  • Lemma 4
  • Definition 5
  • Remark 6
  • Definition 7: Compatibility
  • Definition 8
  • Remark 9
  • Definition 10: Girard posets
  • ...and 17 more