Relative uniform convergence and Archimedean property in pre-ordered vector spaces

Eduard Emelyanov

arxiv: 2602.16419 · v7 · pith:22DYHNYCnew · submitted 2026-02-18 · 🧮 math.FA

Relative uniform convergence and Archimedean property in pre-ordered vector spaces

Eduard Emelyanov This is my paper

Pith reviewed 2026-05-22 11:17 UTC · model grok-4.3

classification 🧮 math.FA

keywords pre-ordered vector spacesrelative uniform topologyArchimedeanizationpositive wedgequotient spaceArchimedean propertyru-topology

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The pith

For any pre-ordered vector space, quotienting by the intersection of the ru-closure of the positive wedge with its negative produces an Archimedeanization.

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

The paper establishes a canonical construction that turns any pre-ordered vector space into an Archimedean one. It defines W as the closure of the positive wedge under the relative uniform topology, sets A equal to the intersection of W with its negative, and shows that the quotient space X/A equipped with the image of W as its positive set is Archimedean. This construction matters for ordered vector space theory because many structural results and representations require the Archimedean property, and the method supplies it directly from the given pre-order via a topological closure step.

Core claim

It is proved that, for a pre-ordered vector space X, the quotient space (X/A,[W]) is an Archimedeanization of X, where W is the closure of the positive wedge X_+ in ru-topology, A=W∩(-W), and [W] is the quotient set of W in X/A.

What carries the argument

The relative uniform topology on X together with the closure W of the positive wedge X_+ and the quotient by A = W ∩ (-W) that carries the induced positive set [W].

If this is right

The quotient space satisfies the Archimedean property by construction.
The vector space operations and the induced order descend to the quotient.
The construction applies to every pre-ordered vector space on which the relative uniform topology is defined.

Where Pith is reading between the lines

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

The same closure-and-quotient pattern may apply to other notions of convergence on ordered spaces.
It supplies a reduction step that lets results proved only for Archimedean spaces be lifted back to the original pre-ordered setting.
The construction could be tested on concrete examples such as spaces of functions equipped with pointwise order.

Load-bearing premise

The relative uniform topology exists on the pre-ordered vector space and the closure W of the positive wedge satisfies the algebraic conditions needed to make the quotient well-defined.

What would settle it

A concrete pre-ordered vector space X in which the quotient space (X/A,[W]) fails to satisfy the Archimedean property or does not serve as the universal target for order-preserving maps into Archimedean spaces.

read the original abstract

It is proved that, for a pre-ordered vector space $X$, the quotient space $(X/A,[W])$ is an Archimedeanization of $X$, where $W$ is the closure of the positive wedge $X_+$ in ru-topology, $A=W\cap(-W)$, and $[W]$ is the quotient set of $W$ in $X/A$.

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

Emelyanov shows that quotienting a pre-ordered vector space by the symmetric part of the ru-closure of its positive wedge yields an Archimedeanization.

read the letter

The main takeaway is that this paper constructs the Archimedeanization of a pre-ordered vector space X explicitly as the quotient (X/A, [W]), where W is the closure of the positive wedge in the relative uniform topology and A is its intersection with the negative. This pins down a concrete way to enforce the Archimedean property through the quotient order. The result is new in the sense that it ties the ru-closure directly to this characterization rather than leaving the construction more abstract. The paper does a good job verifying that W stays a wedge and that the induced order on the quotient satisfies the defining implication: if n[x] ≤ [y] for all natural n, then [x] ≤ 0. The steps follow from the definitions of ru-convergence and closure without obvious circularity or extra assumptions that would break the algebra. The derivations appear solid on the points checked in the stress-test. The main soft spot is the narrow scope. The work stays inside the theory of pre-ordered spaces and does not produce new examples or reach outside ordered functional analysis, so its reach is limited even if the claim holds. No load-bearing flaws show up in the central argument. This is a paper for specialists already comfortable with relative uniform topologies and Archimedean properties in ordered vector spaces. A reader who needs a precise quotient description for Archimedean envelopes will get something usable from it. The grounding in standard definitions and the direct proof make it worth a serious referee's time, even if some polishing on examples could help. I would recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for a pre-ordered vector space X, the quotient space (X/A, [W]) is an Archimedeanization of X, where W is the closure of the positive wedge X_+ in the relative uniform (ru) topology, A = W ∩ (-W), and [W] is the induced quotient wedge in X/A.

Significance. If the result holds, the paper supplies a direct topological construction of the Archimedeanization via ru-closure and quotient, establishing the required algebraic and order properties (including the Archimedean axiom) from the definition of ru-convergence. This strengthens the link between relative uniform topology and order completeness in pre-ordered vector spaces and provides a canonical, non-circular route to the Archimedean hull.

minor comments (3)

[§2] §2: The precise axioms for the ru-topology (e.g., the seminorm or neighborhood base used to define ru-convergence) should be stated explicitly before the closure W is introduced, to make the subsequent verification that W remains a wedge fully self-contained.
The notation [W] for the quotient wedge is introduced without a dedicated definition paragraph; adding one would clarify that [W] is well-defined independently of representatives.
A brief comparison with the classical Archimedeanization (via the quotient by the set of infinitesimals) would help situate the ru-topological construction relative to existing literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. We appreciate the recognition that the result, if established, provides a direct topological construction of the Archimedeanization via ru-closure and quotient.

Circularity Check

0 steps flagged

Derivation is self-contained with no circularity

full rationale

The paper directly constructs the Archimedeanization of a pre-ordered vector space X as the quotient (X/A, [W]), where W is defined as the ru-closure of the positive wedge X_+, A = W ∩ (-W), and [W] is the induced quotient wedge. All required algebraic and order properties, including the Archimedean condition that n[x] ≤ [y] for all n implies [x] ≤ 0, are derived step-by-step from the definitions of relative uniform convergence and topological closure. No steps reduce to fitted inputs, self-definitions, or load-bearing self-citations; the proof is self-contained against the stated axioms of pre-ordered vector spaces and ru-topology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background axioms of vector spaces and pre-orders together with the existence of the relative uniform topology; no free parameters or new entities are introduced in the abstract statement.

axioms (2)

domain assumption X is a vector space equipped with a pre-order whose positive wedge X_+ satisfies the usual compatibility conditions with scalar multiplication and addition.
This is the definition of a pre-ordered vector space used throughout the claim.
domain assumption The relative uniform (ru) topology is a well-defined topology on X in which the closure W of X_+ exists and interacts appropriately with the order.
Invoked implicitly when defining W as the ru-closure; assumed from prior literature on ru-convergence.

pith-pipeline@v0.9.0 · 5577 in / 1412 out tokens · 48385 ms · 2026-05-22T11:17:47.011662+00:00 · methodology

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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/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

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

Theorem 3.3: Let (X, X+) be a POVS, W = (X+)ru, and A = W ∩ (−W). Then (X/A, [W]) is an Archimedeanization of (X, X+).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Aliprantis, R

C.D. Aliprantis, R. Tourky, Cones and Duality. American Mathematical Society, Providence, 2007

work page 2007
[2]

Choi and E.G

M.D. Choi and E.G. Effros, Injectivity and operator spaces. J. Funct. Anal., 24, 156–209, (1977)

work page 1977
[3]

Emelyanov, Archimedean Cones in Vector Spaces

E.Y. Emelyanov, Archimedean Cones in Vector Spaces. J. Conv. Anal., 24, 169–183, (2017)

work page 2017
[4]

Emelyanov, Relative uniform convergence in vector lattices: odds and ends

E. Emelyanov, Relative uniform convergence in vector lattices: odds and ends. J. Math. Sci. 271, 733–742, (2023)

work page 2023
[5]

Emelyanov, N

E. Emelyanov, N. Erkur¸ sun-¨Ozcan, and S. Gorokhova, Collective Order Boundedness of Sets of Operators Between Ordered Vector Spaces. Results Math. 80, 70 (2025)

work page 2025
[6]

Emelyanov, N

E. Emelyanov, N. Erkur¸ sun-¨Ozcan, and S. Gorokhova, Relatively uniformly continuous semigroups on ordered vector spaces. J. Math. Anal. Appl. 560, 130556 (2026)

work page 2026
[7]

Gutman and I.A

A.E. Gutman and I.A. Emelianenkov, Locally convex spaces with all Archimedean cones closed, Sib. Math. J., 64(5), 1117–1136 (2023)

work page 2023
[8]

Ito, On some properties of a vector lattice

T. Ito, On some properties of a vector lattice. Sc. Papers of the College of General Education, Univ. of Tokyo, 17, 161–172, (1967)

work page 1967
[9]

Kadison, A representation theory for commutative topological algebra, Mem

R.V. Kadison, A representation theory for commutative topological algebra, Mem. Amer. Math. Soc., 39, (1951)

work page 1951
[10]

Kalauch and O

A. Kalauch and O. van Gaans, Relatively uniform convergence in partially ordered vector spaaces revisited, Positivity and noncommutative analysis, Trends Math., Birkhauser/Springer, Cham, 2019, 269–280

work page 2019
[11]

Kantorovich, Sur les propri´ et´ es des espaces semi-ordonn´ es lin´ eaires

L.V. Kantorovich, Sur les propri´ et´ es des espaces semi-ordonn´ es lin´ eaires. Comptes Rendus de 1’Acad. Sc. Paris, 202, 813–816, (1936)

work page 1936
[12]

Luxemburg and L.C-Jr

W.A.J. Luxemburg and L.C-Jr. Moore, Archimedean quotient Riesz spaces. Duke Math. J., 34, 725–739, (1967)

work page 1967
[13]

de Pagter and A.W

B. de Pagter and A.W. Wickstead, Free and projective Banach lattices. Proc. R. Soc. Edinb. Sect. A 145, 105–143, (2015)

work page 2015
[14]

Paulsen and M

V.I. Paulsen and M. Tomforde, Vector spaces with an order unit. Indiana Univ. Math. J., 58(3), 1319–1359, (2009)

work page 2009
[15]

Veksler, Archimedean principle in homomorphic images of l-groups and of vector lattices

A.I. Veksler, Archimedean principle in homomorphic images of l-groups and of vector lattices. Izv. Vyshs. Ucebn. Zaved. Matematika, 5(53), 33–38, (1966). 9

work page 1966

[1] [1]

Aliprantis, R

C.D. Aliprantis, R. Tourky, Cones and Duality. American Mathematical Society, Providence, 2007

work page 2007

[2] [2]

Choi and E.G

M.D. Choi and E.G. Effros, Injectivity and operator spaces. J. Funct. Anal., 24, 156–209, (1977)

work page 1977

[3] [3]

Emelyanov, Archimedean Cones in Vector Spaces

E.Y. Emelyanov, Archimedean Cones in Vector Spaces. J. Conv. Anal., 24, 169–183, (2017)

work page 2017

[4] [4]

Emelyanov, Relative uniform convergence in vector lattices: odds and ends

E. Emelyanov, Relative uniform convergence in vector lattices: odds and ends. J. Math. Sci. 271, 733–742, (2023)

work page 2023

[5] [5]

Emelyanov, N

E. Emelyanov, N. Erkur¸ sun-¨Ozcan, and S. Gorokhova, Collective Order Boundedness of Sets of Operators Between Ordered Vector Spaces. Results Math. 80, 70 (2025)

work page 2025

[6] [6]

Emelyanov, N

E. Emelyanov, N. Erkur¸ sun-¨Ozcan, and S. Gorokhova, Relatively uniformly continuous semigroups on ordered vector spaces. J. Math. Anal. Appl. 560, 130556 (2026)

work page 2026

[7] [7]

Gutman and I.A

A.E. Gutman and I.A. Emelianenkov, Locally convex spaces with all Archimedean cones closed, Sib. Math. J., 64(5), 1117–1136 (2023)

work page 2023

[8] [8]

Ito, On some properties of a vector lattice

T. Ito, On some properties of a vector lattice. Sc. Papers of the College of General Education, Univ. of Tokyo, 17, 161–172, (1967)

work page 1967

[9] [9]

Kadison, A representation theory for commutative topological algebra, Mem

R.V. Kadison, A representation theory for commutative topological algebra, Mem. Amer. Math. Soc., 39, (1951)

work page 1951

[10] [10]

Kalauch and O

A. Kalauch and O. van Gaans, Relatively uniform convergence in partially ordered vector spaaces revisited, Positivity and noncommutative analysis, Trends Math., Birkhauser/Springer, Cham, 2019, 269–280

work page 2019

[11] [11]

Kantorovich, Sur les propri´ et´ es des espaces semi-ordonn´ es lin´ eaires

L.V. Kantorovich, Sur les propri´ et´ es des espaces semi-ordonn´ es lin´ eaires. Comptes Rendus de 1’Acad. Sc. Paris, 202, 813–816, (1936)

work page 1936

[12] [12]

Luxemburg and L.C-Jr

W.A.J. Luxemburg and L.C-Jr. Moore, Archimedean quotient Riesz spaces. Duke Math. J., 34, 725–739, (1967)

work page 1967

[13] [13]

de Pagter and A.W

B. de Pagter and A.W. Wickstead, Free and projective Banach lattices. Proc. R. Soc. Edinb. Sect. A 145, 105–143, (2015)

work page 2015

[14] [14]

Paulsen and M

V.I. Paulsen and M. Tomforde, Vector spaces with an order unit. Indiana Univ. Math. J., 58(3), 1319–1359, (2009)

work page 2009

[15] [15]

Veksler, Archimedean principle in homomorphic images of l-groups and of vector lattices

A.I. Veksler, Archimedean principle in homomorphic images of l-groups and of vector lattices. Izv. Vyshs. Ucebn. Zaved. Matematika, 5(53), 33–38, (1966). 9

work page 1966