Quantales carrying ortholattice structure

TL;DR

This paper explores the intersection of residuated structures and orthomodular lattices, demonstrating that certain non-Boolean structures can unify quantum and many-valued logic reasoning.

Contribution

It characterizes Girard posets with inversions and constructs a non-Boolean orthomodular structure on the lattice of closed subspaces of real space.

Findings

Complemented lattices with integral residuated structures are Boolean.

The lattice of closed subspaces of real space admits an orthomodular and Girard quantale 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 $C(\mathbb{R}^n)$ 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.