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

Two complementary perspectives have emerged in this endeavor. The first, rooted in the work of Birkhoff and von Neumann, employs complete orthomodular lattices to formalize the testable properties of quantum systems. These lattices provide a static, algebraic foundation for quantum logic, encoding the structure of propositions and their orthogonality relations. The second perspective, developed more recently through dynamic epistemic logic and quantum dynamic algebras, emphasizes the operational and transformational aspects of quantum theory-how quantum actions compose, interact, and modify the state of a system.

The dynamic approach to quantum logic was pioneered by [1]. They introduced quantum dynamic logic to reason about measurements, operations, and information flow, extending classical dynamic logic to the non-Boolean quantum setting. Building on this foundation, [5] developed the theory of orthomodular dynamic algebras, providing an algebraic counterpart to Baltag and Smets’ logical framework and establishing initial connections to orthomodular lattices.

This paper establishes a precise categorical equivalence between these two viewpoints. Specifically, we demonstrate that the category of complete orthomodular lattices is equivalent to the category of T -based orthomodular dynamic algebras, a specialized class of unital involutive quantales designed to formalize quantum actions. The equivalence shows that the static lattice-theoretic view and the dynamic, quantale-based view are categorically equivalent, providing a direct bridge between these perspectives.

Our approach builds upon and refines the work of [5], which established initial links between orthomodular lattices and dynamic algebras. We extend their results by introducing a more flexible framework based on involutive submonoids of the Foulis quantale of linear maps. For a complete orthomodular lattice M = (M, ≤, (-) ⊥ ), we construct an involutive submonoid L M satisfying

where π m denotes the Sasaki projection onto m and Lin(M) is the Foulis quantale of linear endomaps on M. This intermediate choice between Sasaki projections and all linear maps provides fine-grained control over the resulting dynamic algebra structure while maintaining full compatibility with the orthomodular lattice operations.

The free unital involutive quantale P(L M ) of all subsets of L M , equipped with setwise composition and an appropriately defined orthocomplement operator ∼, yields a T -based orthomodular dynamic algebra whose “test set” is naturally identified with the elements of the original lattice M. This construction is functorial, and we establish that the resulting functors between the categories COL and T ODA form an equivalence via natural isomorphisms in both directions. In doing so, we clarify how dynamics emerge from orthomodular lattices and strengthen the connection with unital involutive quantales, providing a comprehensive algebraic foundation for the dynamic quantum logic initiated by Baltag and Smets.

The significance of this equivalence extends beyond pure category theory. Complete orthomodular lattices encompass several important classes of quantum-logical structures, including Hilbert lattices-the lattices of closed subspaces of Hilbert spaces that arise directly from the mathematical formalism of quantum mechanics. Our equivalence therefore provides a robust categorical link between quantales and a broad spectrum of quantum-theoretic frameworks, from abstract quantum logic to concrete operator algebras. This connection facilitates the transfer of results, techniques, and intuitions between these traditionally distinct areas of study, and demonstrates that the dynamic perspective on quantum logic advocated by Baltag and Smets is not merely a convenient formalism but is categorically equivalent to the classical static perspective.

Methodologically, our contribution lies in providing an explicit, constructive proof of this equivalence that emphasizes conceptual clarity and computational feasibility. By working directly within the Foulis quantale Lin(M) and carefully controlling the generator scheme L M , we obtain a streamlined demonstration of minimality and a transparent interaction between the orthocomplement operator ∼ and Sasaki projections. The approach simplifies prior arguments and facilitates computational implementation and extensions to broader structures.

The paper is organized as follows. Section 2 reviews the necessary background on quantales, involutive structures, Foulis quantales, and complete orthomodular lattices, establishing notation and recalling essential results. Section 3 i