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.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.

arXiv:2602.17273Feb 19, 20260 citations

Orthomodular structures as quantum logics

Pavel Pták, Sylvia Pulmannová

Kluwer Academic Publishers eBooksJan 1, 1991432 citations

Categorical Equivalence Between Finitary Orthomodular Dynamic Algebras and Orthomodular Lattices

Juanda Kelana Putra, Richard Smolka

This paper reveals a categorical equivalence connecting two distinct quantum logic structures. The first is the orthomodular lattice, an algebraic system designed to formalize the properties of quantum systems. The second is a finitary orthomodular dynamic algebra, a specialized development of the orthomodular dynamic algebra where the underlying quantum actions are restricted to be finitary. The applicability of the result extends to more specialized lattices, such as Hilbert lattices of closed subspaces of a Hilbert space, beyond general orthomodular lattices. As these lattice structures exhibit connections to a diverse array of quantum structures, the established equivalence categorically bridges unital involutive m-semilattices with a broad spectrum of quantum formalisms.This paper reveals a categorical equivalence connecting two distinct quantum logic structures. The first is the orthomodular lattice, an algebraic system designed to formalize the properties of quantum systems. The second is a finitary orthomodular dynamic algebra, a specialized development of the orthomodular dynamic algebra where the underlying quantum actions are restricted to be finitary. The applicability of the result extends to more specialized lattices, such as Hilbert lattices of closed subspaces of a Hilbert space, beyond general orthomodular lattices. As these lattice structures exhibit connections to a diverse array of quantum structures, the established equivalence categorically bridges unital involutive m-semilattices with a broad spectrum of quantum formalisms.

arXiv:2604.17004Apr 18, 20260 citations

Subalgebras of Orthomodular Lattices

John Harding, Mirko Navara

OrderDec 30, 201025 citations

\mathscr{T}

-based orthomodular dynamic algebras

This paper establishes a categorical equivalence between the category

\mathbb{COL}

of complete orthomodular lattices and the category

\mathscr{T}\mathbb{ODA}

\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.