From orthoposets to orthomodular posets

Bert Lindenhovius; Gejza Jenda; John Harding

arxiv: 2607.01259 · v1 · pith:QCS25K3Fnew · submitted 2026-06-09 · 🧮 math.CO · math-ph· math.CT· math.MP· quant-ph

From orthoposets to orthomodular posets

John Harding , Gejza Jenda , Bert Lindenhovius This is my paper

Pith reviewed 2026-07-03 23:46 UTC · model grok-4.3

classification 🧮 math.CO math-phmath.CTmath.MPquant-ph

keywords orthoposetsorthomodular posetsstrong orthoposetscoreflective subcategoryortholatticesorthocomplementationorder modification

0 comments

The pith

The category of orthomodular posets is a full coreflective subcategory of strong orthoposets obtained by modifying the order on any strong orthoposet.

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

The paper establishes a concrete construction that takes any strong orthoposet and produces an orthomodular poset on exactly the same set with the same orthocomplementation but a redefined partial order. This yields a functor that is a coreflector, so the inclusion of orthomodular posets into strong orthoposets is a full coreflective embedding. The same construction restricts to a right adjoint functor from the category of ortholattices into orthomodular posets. A reader would care because it supplies an explicit way to associate to each member of the larger class a canonical member of the more restrictive class while preserving the underlying set and complementation operation.

Core claim

We show that the category of orthomodular posets is a full coreflective subcategory of the category of strong orthoposets, those orthoposets in which any two orthogonal elements have a join. This coreflection is obtained by building from a strong orthoposet P, an orthomodular poset with the same underlying set and same orthocomplementation as P, but with modified order. This coreflector restricts to a functor from the category of ortholattices to the category of orthomodular posets, and this functor is right adjoint.

What carries the argument

The order-modification construction that turns a strong orthoposet into an orthomodular poset on the identical set while preserving orthocomplementation.

If this is right

Every strong orthoposet determines a unique orthomodular poset with the same elements and orthocomplementation.
The inclusion of orthomodular posets into strong orthoposets has a right adjoint given by this order modification.
The construction restricts to ortholattices and remains right adjoint in that setting.
Strong orthoposets already satisfying the orthomodular law are fixed by the construction.

Where Pith is reading between the lines

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

The difference between the original and modified orders on a given strong orthoposet may encode a canonical measure of how far the structure is from being orthomodular.
One could ask whether this coreflection preserves or reflects completeness or other lattice-theoretic properties when they are present.
The same order-modification idea might be tested on still larger classes such as general orthoposets without the strong-join assumption.

Load-bearing premise

The particular redefinition of the partial order on a strong orthoposet produces a structure obeying the orthomodular law.

What would settle it

An explicit strong orthoposet together with the proposed order change for which the resulting poset fails the orthomodular law or the adjunction identities fail to hold.

read the original abstract

We show that the category of orthomodular posets is a full coreflective subcategory of the category of strong orthoposets, those orthoposets in which any two orthogonal elements have a join. This coreflection is obtained by building from a strong orthoposet $P$, an orthomodular poset with the same underlying set and same orthocomplementation as $P$, but with modified order. This coreflector restricts to a functor from the category of ortholattices to the category of orthomodular posets, and this functor is right adjoint.

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 constructs a coreflector from strong orthoposets to orthomodular posets by modifying the order on the same set, giving a new categorical relationship with an adjoint on ortholattices.

read the letter

The main takeaway is that orthomodular posets form a full coreflective subcategory of strong orthoposets through a specific order modification on the same set and orthocomplementation.

The paper constructs this coreflector and shows it restricts to a right adjoint from ortholattices. This is new as a categorical relationship and does a good job of linking these structures with adjoint functors that can transfer results between categories.

The construction itself is the positive part: it keeps the underlying set and complementation fixed while changing the order to enforce the orthomodular law. If the modification is spelled out clearly in the paper, this gives a concrete way to associate an orthomodular poset to any strong orthoposet.

The potential issue is whether the new partial order is well-defined and satisfies the orthomodular identity in all cases. The abstract states that it does, but without the explicit relation or the proof that it works for orthogonal elements and comparable pairs, the claim rests on that step. If the paper provides the definition and verifies it, the result holds; otherwise the coreflection fails. The stress-test concern about the modification producing the orthomodular law is the right one to check first.

This is for researchers in quantum logic and order-theoretic approaches to it. A reader who works with orthoposets and wants categorical tools would find the adjunctions of interest. It is worth sending to peer review so that the construction can be checked in detail by experts in the area.

Referee Report

3 major / 2 minor

Summary. The manuscript proves that the category of orthomodular posets (OMP) is a full coreflective subcategory of the category of strong orthoposets (SOP). The coreflection is realized by a construction that, given a strong orthoposet P, produces an orthomodular poset on the same underlying set with the same orthocomplementation but with a modified partial order; the resulting functor is shown to be a right adjoint to the inclusion, and the construction restricts to a right adjoint from the category of ortholattices to OMP.

Significance. If the central construction is correct, the result supplies an explicit, set-preserving way to associate an orthomodular poset to any strong orthoposet while retaining the orthocomplementation. This yields a categorical relationship between the two classes of structures that is potentially useful for studying quantum logics and orthomodular posets. The restriction to ortholattices and the adjunction property are additional structural features that strengthen the contribution.

major comments (3)

[§3] §3 (Construction of the modified order): the explicit definition of the new relation ≤' on the carrier of a strong orthoposet P must be stated precisely (including how it differs from the original ≤ while preserving orthogonality and the orthocomplement). Without this formula it is impossible to verify that (P, ≤', ') satisfies the orthomodular law a ∨ (a' ∧ b) = b whenever a ≤' b.
[Theorem 4.1] Theorem 4.1 (Coreflection): the proof that the modified structure is orthomodular and that the inclusion OMP ↪ SOP is a full coreflector requires an explicit check that the new order is a partial order, that all joins of orthogonal pairs remain compatible, and that the unit of the adjunction satisfies the universal property for every strong orthoposet.
[§5] §5 (Restriction to ortholattices and right-adjoint property): the claim that the coreflector restricts to a right adjoint from ortholattices to OMP must be accompanied by a verification that the hom-set bijection holds when the domain is restricted to ortholattices; the current sketch does not address whether the modified order preserves the lattice operations.

minor comments (2)

[Notation] Ensure that every occurrence of the original order ≤ and the modified order ≤' is typographically distinguished throughout the proofs.
[Preliminaries] Add a short diagram or table comparing the axioms satisfied by strong orthoposets versus orthomodular posets to clarify the difference the construction is intended to bridge.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where greater explicitness will strengthen the manuscript. We address each major comment below and will revise the paper to incorporate the requested clarifications and verifications.

read point-by-point responses

Referee: [§3] §3 (Construction of the modified order): the explicit definition of the new relation ≤' on the carrier of a strong orthoposet P must be stated precisely (including how it differs from the original ≤ while preserving orthogonality and the orthocomplement). Without this formula it is impossible to verify that (P, ≤', ') satisfies the orthomodular law a ∨ (a' ∧ b) = b whenever a ≤' b.

Authors: We agree that the definition of the modified order ≤' requires a more formal and self-contained statement. In the revised manuscript we will insert an explicit formula for a ≤' b (distinct from the original ≤, while preserving the orthocomplement and orthogonality relation) and will immediately verify that the resulting structure satisfies the orthomodular law whenever a ≤' b. revision: yes
Referee: [Theorem 4.1] Theorem 4.1 (Coreflection): the proof that the modified structure is orthomodular and that the inclusion OMP ↪ SOP is a full coreflector requires an explicit check that the new order is a partial order, that all joins of orthogonal pairs remain compatible, and that the unit of the adjunction satisfies the universal property for every strong orthoposet.

Authors: We will expand the proof of Theorem 4.1 to supply the missing explicit verifications: that ≤' is a partial order, that joins of orthogonal pairs remain compatible under the new order, and that the unit of the adjunction satisfies the universal property with respect to every strong orthoposet. revision: yes
Referee: [§5] §5 (Restriction to ortholattices and right-adjoint property): the claim that the coreflector restricts to a right adjoint from ortholattices to OMP must be accompanied by a verification that the hom-set bijection holds when the domain is restricted to ortholattices; the current sketch does not address whether the modified order preserves the lattice operations.

Authors: We will augment §5 with a direct verification of the hom-set bijection under the restriction to ortholattices and will clarify the interaction between the modified order and the existing lattice operations, confirming that the right-adjoint property is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: pure category-theoretic construction

full rationale

The paper is a self-contained existence theorem in category theory. It defines strong orthoposets, constructs an explicit new partial order on the same carrier set with the same orthocomplementation, and verifies that the resulting structure is an orthomodular poset, yielding a coreflector. No parameters are fitted, no predictions are made from data, no self-citations are load-bearing for the central claim, and no ansatz or uniqueness theorem is smuggled in. The derivation consists of direct algebraic verification of the orthomodular law under the modified order, which is independent of the target result. This is the normal case of a non-circular pure-math construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no free parameters, invented entities, or non-standard axioms are mentioned.

axioms (1)

standard math Standard definitions and axioms of posets, orthocomplementation, orthomodular posets, and coreflective subcategories in category theory.
The result relies on these background definitions from order theory and category theory.

pith-pipeline@v0.9.1-grok · 5628 in / 1394 out tokens · 27939 ms · 2026-07-03T23:46:50.445297+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Kalmbach,Orthomodular Lattices, London Mathematical Society Monographs, Vol

G. Kalmbach,Orthomodular Lattices, London Mathematical Society Monographs, Vol. 18, Academic Press, London, 1983

work page 1983
[2]

Pt´ ak and S

P. Pt´ ak and S. Pulmannov´ a,Orthomodular Structures as Quantum Logics: Intrinsic Properties, State Space and Probabilistic Topics, Fundamental Theories of Physics, Vol. 44, Kluwer Academic Publishers, Dordrecht, 1991. New Mexico State University, Las Cruces NM 88003, USA Email address:jharding@nmsu.edu Department of Mathematics and Descriptive Geometry,...

work page 1991

[1] [1]

Kalmbach,Orthomodular Lattices, London Mathematical Society Monographs, Vol

G. Kalmbach,Orthomodular Lattices, London Mathematical Society Monographs, Vol. 18, Academic Press, London, 1983

work page 1983

[2] [2]

Pt´ ak and S

P. Pt´ ak and S. Pulmannov´ a,Orthomodular Structures as Quantum Logics: Intrinsic Properties, State Space and Probabilistic Topics, Fundamental Theories of Physics, Vol. 44, Kluwer Academic Publishers, Dordrecht, 1991. New Mexico State University, Las Cruces NM 88003, USA Email address:jharding@nmsu.edu Department of Mathematics and Descriptive Geometry,...

work page 1991