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On $\mathscr{T}$-based orthomodular dynamic algebras

Jan Paseka, Juanda Kelana Putra, Richard Smolka

Abstract

This paper establishes a categorical equivalence between the category $\mathbb{COL}$ of complete orthomodular lattices and the category $\mathscr{T}\mathbb{ODA}$ of $\mathscr{T}$-based orthomodular dynamic algebras. Complete orthomodular lattices serve as the static algebraic foundation for quantum logic, modeling the testable properties of quantum systems. In contrast, $\mathcal{T}$-based orthomodular dynamic algebras, which are specialized unital involutive quantales, formalize the composition and quantum-logical properties of quantum actions. This result refines prior connections between orthomodular lattices and dynamic algebras, provides a constructive bridge between static and dynamic quantum logic perspectives, and extends naturally to Hilbert lattices and broader quantum-theoretic structures.

On $\mathscr{T}$-based orthomodular dynamic algebras

Abstract

This paper establishes a categorical equivalence between the category of complete orthomodular lattices and the category of -based orthomodular dynamic algebras. Complete orthomodular lattices serve as the static algebraic foundation for quantum logic, modeling the testable properties of quantum systems. In contrast, -based orthomodular dynamic algebras, which are specialized unital involutive quantales, formalize the composition and quantum-logical properties of quantum actions. This result refines prior connections between orthomodular lattices and dynamic algebras, provides a constructive bridge between static and dynamic quantum logic perspectives, and extends naturally to Hilbert lattices and broader quantum-theoretic structures.
Paper Structure (16 sections, 27 theorems, 155 equations)

This paper contains 16 sections, 27 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantales and related structures
  4. The category $\mathbb{COL}$ of complete orthomodular lattices
  5. Involutive generalized dynamic algebras
  6. The category $\mathbb{\mathscr{T}ODA}$ of $\mathscr{T}$-based orthomodular dynamic algebras
  7. $\mathscr{T}$-based orthomodular dynamic algebras
  8. A concrete choice of the functor $\mathscr{T}$
  9. Construction of $\mathscr{T}$-based orthomodular dynamic algebras from complete orthomodular lattices
  10. Mapping of Objects
  11. Mapping of Arrows
  12. The Functor $\Psi$ from $\mathscr{T}$-based orthomodular dynamic algebras to complete orthomodular lattices
  13. Mapping of Objects
  14. Mapping of Arrows
  15. The Natural Isomorphism $\mu : 1_{\mathbb{COL}} \stackrel{\cong}{\Rightarrow} \Psi\circ \Gamma$
  16. ...and 1 more sections

Key Result

Proposition 2.7

A unital involutive quantale $\mathcal{Q}=(Q,\bigsqcup, \odot, ^*, e)$ is a Foulis quantale with an endomap $[\,-\,] \colon Q\rightarrow Q$ if and only if there is an endomap $-\!^{\perp}\colon Q\to Q$ satisfying the following conditions for all $s,x \in Q$: where $s\perp x$ iff $s^{*}\odot x=0$.

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • Remark 2.9
  • ...and 62 more