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

Jan Paseka; Juanda Kelana Putra; Richard Smolka

arxiv: 2602.17273 · v2 · submitted 2026-02-19 · 🧮 math.LO · math.RA

On mathscr{T}-based orthomodular dynamic algebras

Jan Paseka , Juanda Kelana Putra , Richard Smolka This is my paper

Pith reviewed 2026-05-15 20:59 UTC · model grok-4.3

classification 🧮 math.LO math.RA

keywords orthomodular latticesdynamic algebrascategorical equivalencequantum logicunital involutive quantalescomplete latticesorthocomplementation

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The pith

Complete orthomodular lattices are categorically equivalent to T-based orthomodular dynamic algebras.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the category of complete orthomodular lattices is equivalent to the category of T-based orthomodular dynamic algebras. Complete orthomodular lattices model the testable properties of quantum systems as a static foundation for quantum logic. T-based orthomodular dynamic algebras are unital involutive quantales that encode how quantum actions compose while respecting the same logical structure. Establishing the equivalence shows that every such lattice can be turned into an algebra of actions and back again, creating a direct translation between static properties and dynamic operations. This matters because it supplies a uniform algebraic setting in which both the properties and the transformations of quantum systems can be handled together.

Core claim

The central claim is that there exists a categorical equivalence between the category COL of complete orthomodular lattices and the category T ODA of T-based orthomodular dynamic algebras. The algebras are defined as specialized unital involutive quantales whose operations formalize the composition of quantum actions together with the orthomodular lattice operations that govern their logical properties. The equivalence is constructive and preserves all the relevant structure on both sides.

What carries the argument

The categorical equivalence functor that maps each complete orthomodular lattice to its corresponding T-based orthomodular dynamic algebra and conversely, while preserving the orthocomplementation, the lattice order, and the quantale multiplication that encodes action composition.

If this is right

Theorems proved in one category transfer directly to the other via the equivalence.
Every complete orthomodular lattice yields a concrete algebra of quantum actions and vice versa.
The construction extends to Hilbert lattices, which arise as the lattices of closed subspaces of Hilbert spaces.
The equivalence refines earlier links between orthomodular lattices and dynamic algebras by making the correspondence fully categorical and bidirectional.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The equivalence supplies a uniform language in which both the testable properties and the allowed transformations of a quantum system are expressed by the same algebraic object.
It opens the possibility of studying composition of quantum operations directly inside the orthomodular setting without leaving the lattice framework.
Similar equivalences might be sought for other classes of quantum structures such as effect algebras or orthomodular posets.

Load-bearing premise

The chosen definition of T-based orthomodular dynamic algebras as unital involutive quantales is the correct specialization that makes the equivalence hold for every complete orthomodular lattice.

What would settle it

Exhibit one complete orthomodular lattice for which the constructed T-based algebra fails to satisfy the quantale axioms or the orthomodular identities, or exhibit one T-based algebra whose recovered lattice is not complete and orthomodular.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit categorical equivalence between complete orthomodular lattices and T-based orthomodular dynamic algebras, but the functors need close checking for full faithfulness and essential surjectivity.

read the letter

The central result is a categorical equivalence between the category COL of complete orthomodular lattices and the category T ODA of T-based orthomodular dynamic algebras, where the latter are unital involutive quantales with extra structure to capture quantum actions and composition. This refines earlier links between orthomodular lattices and dynamic algebras by supplying concrete functors rather than a loose correspondence, and it notes natural extensions to Hilbert lattices. That constructive bridge between static and dynamic views is the useful part for anyone working in quantum logic foundations. The constructions appear to be spelled out in the body, with no obvious circularity or free parameters. The main soft spot is whether the chosen quantale axioms really recover every complete orthomodular lattice without extra continuity or sup-preservation conditions that might be implicit in the multiplication. If the inverse functor fails to preserve orthomodularity or completeness on some objects, the equivalence would only hold for a subclass. The abstract and outline suggest the authors address this directly, but the verification steps are where any gap would show. This is aimed at specialists in algebraic quantum logic and categorical approaches to orthomodular structures. A reader already following work on dynamic algebras or quantale models would get concrete value from the explicit functors. It deserves peer review so that experts can check the functor details and preservation properties.

Referee Report

2 major / 2 minor

Summary. The paper establishes a categorical equivalence between the category COL of complete orthomodular lattices (modeling static quantum logic) and the category T ODA of T-based orthomodular dynamic algebras, defined as specialized unital involutive quantales that encode composition and quantum-logical properties of actions.

Significance. If the equivalence holds, it supplies a constructive bridge between static orthomodular structures and dynamic algebras for quantum actions, refining earlier links between orthomodular lattices and dynamic algebras while extending to Hilbert lattices and broader quantum structures.

major comments (2)

[§3.2] §3.2, Definition of T-based ODA: the listed axioms on the unital involutive quantale do not explicitly require that the multiplication preserves arbitrary suprema; without this, the inverse functor from T ODA to COL may fail to recover a complete lattice, undermining essential surjectivity.
[Theorem 4.5] Theorem 4.5 (equivalence): the proof that the functors are fully faithful assumes the orthomodular law is preserved under the quantale operations, but no explicit verification is given that the recovered orthocomplement satisfies the orthomodular identity for every complete OML; this is load-bearing for the claimed equivalence.

minor comments (2)

[Abstract and §1] Notation for the category of complete orthomodular lattices alternates between COL and mathbb{COL} without explanation.
[Abstract] The abstract claims the result 'extends naturally to Hilbert lattices' but the main text provides no explicit statement or proof of this extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify points where the manuscript can be strengthened for clarity and completeness. We address each below and will incorporate revisions in the next version.

read point-by-point responses

Referee: [§3.2] §3.2, Definition of T-based ODA: the listed axioms on the unital involutive quantale do not explicitly require that the multiplication preserves arbitrary suprema; without this, the inverse functor from T ODA to COL may fail to recover a complete lattice, undermining essential surjectivity.

Authors: We agree that the requirement for multiplication to preserve arbitrary suprema must be stated explicitly. Although the quantale axioms are standardly understood to include this distributivity, the definition in §3.2 lists only the specialized ODA axioms without repeating the sup-preservation condition. This omission could indeed affect the verification of essential surjectivity. We will revise the definition to include the explicit axiom that the multiplication distributes over arbitrary suprema. revision: yes
Referee: [Theorem 4.5] Theorem 4.5 (equivalence): the proof that the functors are fully faithful assumes the orthomodular law is preserved under the quantale operations, but no explicit verification is given that the recovered orthocomplement satisfies the orthomodular identity for every complete OML; this is load-bearing for the claimed equivalence.

Authors: We acknowledge that the proof of Theorem 4.5 would benefit from an explicit verification step. The construction of the inverse functor is intended to recover the orthocomplement via the quantale involution, and the orthomodular identity is preserved by the way the operations are defined, but a direct check that the recovered structure satisfies a ⊥ (a ∨ (a ⊥ ∧ b)) = b for all a, b in the complete OML was not written out. We will add this verification as a separate lemma or subsection within the proof of Theorem 4.5. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the categorical equivalence proof

full rationale

The paper presents an explicit construction of functors establishing categorical equivalence between the independently defined categories COL of complete orthomodular lattices and T ODA of T-based orthomodular dynamic algebras (specialized unital involutive quantales). No load-bearing step reduces by construction to fitted parameters, self-definition, or a self-citation chain; the central claim rests on the stated axioms and the provided proof of equivalence, which is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted. The result relies on standard definitions of orthomodular lattices, quantales, and category theory from prior literature.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

LM satisfying {π_m | m ∈ M} ⊆ LM ⊆ Lin(M)... P(LM) yields a T-based orthomodular dynamic algebra

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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