Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.17004

Categorical Equivalence Between Finitary Orthomodular Dynamic Algebras and Orthomodular Lattices

Juanda Kelana Putra, Richard Smolka

Published Apr 18, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2604.17004

Related papers

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

Jan Paseka, Juanda Kelana Putra, Richard Smolka

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:2602.17273•Feb 19, 2026•0 citations

Sublattices and Δ-blocks of orthomodular posets

Ivan Chajda, Helmut Länger

Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation $\mathrel \Delta$ and a binary operator $d(x,y)$ that are generalizations of the binary relation $\textrm{C}$ and the commutator $c(x,y)$, respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations $\mathrel \Delta$ and $\leftrightarrow$ (the latter was introduced by P. Pták and S. Pulmannová) and the operator $d(x,y)$. In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of $\Delta$-blocks with respect to Boolean subalgebras and distributive subposets they include.

arXiv:2004.01962•Journal of Logic and Computation•Jun 25, 2020•0 citations

Quantifiers for quantum logic

Chris Heunen

We consider categorical logic on the category of Hilbert spaces. More generally, in fact, any pre-Hilbert category suffices. We characterise closed subobjects, and prove that they form orthomodular lattices. This shows that quantum logic is just an incarnation of categorical logic, enabling us to establish an existential quantifier for quantum logic, and conclude that there cannot be a universal quantifier.

arXiv:0811.1457•Nov 10, 2008•2 citations

States and Boolean Algebra

Nabila Nassiief Mikhaeel, Basim Samir Labib

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

Algebra Colloquium•Dec 1, 2008•0 citations

Posibilidades de generalización de las lógicas cuánticas

Carlos Lungarzo

The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.) LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes. LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. In § 2 we use infinitarian logics, where a numerable number of conjunctions and quantificators is permitted, to express the theory of orthomodular partially ordered sets (posets). That formulation uses essentially the logic called (omega-1, omega-1), and a conjecture is raised about proving it cannot be realized within a weaker logic. The way to prove this, it is suggested, should be analogous to the way the indefinability of well-ordering is proved. Such generalizations are analogous to those obtained when we pass, e.g., from the field of real numbers to any commutative, Archimedian, etc., field; then to a commutative ring with and unity, and so on. They are, therefore, formal, in the sense that (1) they do not “say” much about the initial structure which motivated the generalization and that (2) in generalizing, we allow the “entrance” of new structures that have little to do with the original motivation. For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense. In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure. A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected. Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates. Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all. It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5. Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2. In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces. For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.) As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space. Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics. Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness. [C.L.]

Crítica (México D F En línea)•Nov 9, 1979•0 citations

Systems of quantum logic

R I G Hughes

According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.

•Jan 1, 2010•0 citations
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.
On T\mathscr{T}T-based orthomodular dynamic algebras
This paper establishes a categorical equivalence between the category COL\mathbb{COL}COL of complete orthomodular lattices and the category TODA\mathscr{T}\mathbb{ODA}TODA of T\mathscr{T}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, T\mathcal{T}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.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation Δ\mathrel \DeltaΔ and a binary operator d(x,y)d(x,y)d(x,y) that are generalizations of the binary relation C\textrm{C}C and the commutator c(x,y)c(x,y)c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ\mathrel \DeltaΔ and ↔\leftrightarrow↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y)d(x,y)d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Δ\DeltaΔ-blocks with respect to Boolean subalgebras and distributive subposets they include.
The complete and atomic orthomodular lattice structure which is formed by the projectors (or equivalently, by the closed subspaces) of a complex and separable Hilbert space will be indicated in this paper by LG (H), where H is the Hilbert space. (We take operations and distinct elements for granted.) LG (H) is, classically considered, the “typical” logic of quantum mechanics, in the following sense: if H is the space associated with physical system S, then H’s rays, the ultramaximal operators, etc., represent typical properties of S, such as observable states and magnitudes. LG (H) is simply an element in model sets of more general theories such as the orthomodular lattice set, which we formalize in § 2, using a second-order logic. This generalization, which we call “formal” because it consists merely of giving more permissive axioms that facilitate the apparition of new structures other than LG (H), can yet be extended. In § 2 we use infinitarian logics, where a numerable number of conjunctions and quantificators is permitted, to express the theory of orthomodular partially ordered sets (posets). That formulation uses essentially the logic called (omega-1, omega-1), and a conjecture is raised about proving it cannot be realized within a weaker logic. The way to prove this, it is suggested, should be analogous to the way the indefinability of well-ordering is proved. Such generalizations are analogous to those obtained when we pass, e.g., from the field of real numbers to any commutative, Archimedian, etc., field; then to a commutative ring with and unity, and so on. They are, therefore, formal, in the sense that (1) they do not “say” much about the initial structure which motivated the generalization and that (2) in generalizing, we allow the “entrance” of new structures that have little to do with the original motivation. For instance, in the theory of orthomodular lattices, besides Hilbert-space logics (or those based on other topological vectorial spaces), new models appear, such as finite or orthomodular lattices, for which the name “quantum logic” has little sense. In § 3 we study the modifications, and eventual generalizations, of the basic structure that generates quantum logic, i.e., the infinite-dimensional separable complex Hilbert space structure. A first way seems to be the generalizing of the field, and it is analyzed in § 3.1. In principle, it seems natural to admit that the field must contain some copy of the field of real numbers, or have at least an expressive force as great as theirs. This is why the use of real fields —and for a greater certainty, of complex fields— in the H spaces of quantum mechanics seems to be fully justified. They study of quaternions, carried on by Emch and others, but not clearly completed yet, could lead to an equivalent quantum mechanics, notwithstanding the non-commutativity of this field. Anyway, real, complex and quaternions seem to be only fields with a “natural” right to sustentate these spaces. The incompleteness of rational fields, and their undesirable properties as to dimension and measurability, cause them to be rejected. Other proposals are studied. Leaving aside Hensel’s bodies, that seem to be wholly different in nature, we are left with p-adics Qp and Galois’ fields as the only reasonable candidates. Know algebraic results are then invoked, according to which an extension or completion of these bodies would not lead to the desired results. Quite simply, we would not, save on trivial instances, have any logic at all. It is a well-known fact that by completing p-adic fields (let K be such a completion) we arrive at space (V, K) upon that field, which does not allow to define hermitian forms for dimensions greater than 4. Actually, no involutory anti-automorphism of the field (such as complex conjugation) permits to associate any quadratic form for a dimension greater than or equal to 5. Things are even worse in the case of Galois’ fields where limitative results begin to appear already in dimensions greater than 2. In § 3.2 we choose to accept the complex numbers body C as a canonic body, and see whether it is possible to generalize space into a class that need not be Hilbert’s. There are various possibilities: to consider Banach spaces, or else more or less general local convex spaces, and particularly, to focus analysis on Mackey spaces. For Banach spaces there appear Maczynski’s limitative results; for Mackey spaces, “analogous” results from Wilbur. (Cf. references at the end of the essay.) As a matter of fact, only infinite-dimensional spaces can be of any interest. In this case, if a Mackey space has an “acceptable” topology, i.e., a stronger one than the so-called weak convergence topology, and its LG (M) is really a logic, then M is once again a Hilbert space. Several of these results to the conclusion that we can hardly obtain a different logic (a more general or anyway more informative one) by taking the class LG (E) of some non-Hilbert space E. Rayski’s proposition about substituting Schwartz space for Hilbert space is also considered, but although such a substitution would modify actual work “within” the space in the interests of formalization, it is natural to expect the apparition of a logic LG (S), equivalent to former logics. Maybe the part special logics play in quantum mechanics will be entirely unnecessary when the axiomatizing of a physical system can be done in terms of some kind of spaces that leaves no doubt about its usefulness. [C.L.]
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.