A note on disjointness and discrete elements in partially ordered vector spaces

Jani Jokela

arxiv: 2509.11212 · v4 · submitted 2025-09-14 · 🧮 math.FA

A note on disjointness and discrete elements in partially ordered vector spaces

Jani Jokela This is my paper

Pith reviewed 2026-05-18 16:47 UTC · model grok-4.3

classification 🧮 math.FA

keywords D-disjointnessD-discrete elementspre-Riesz spacespervasivenessweak pervasivenessArchimedean ordered spacespartially ordered vector spacesdisjointness generalizations

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The pith

In finite-dimensional Archimedean pre-Riesz spaces, pervasiveness and weak pervasiveness are equivalent.

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

The paper examines three generalizations of disjointness from vector lattices to partially ordered vector spaces, with D-disjointness as the most general. It develops basic properties of D-discrete elements in Archimedean spaces and connects them to discrete elements in pre-Riesz theory. The central result shows that pervasiveness and weak pervasiveness coincide exactly when the space is finite-dimensional and Archimedean. A sympathetic reader would care because this equivalence reduces two order-theoretic notions to one in a broad class of spaces that properly contains vector lattices.

Core claim

The notions of pervasiveness and weak pervasiveness are equivalent in every finite-dimensional Archimedean pre-Riesz space. This follows from establishing properties of D-discrete elements in Archimedean partially ordered spaces and relating D-disjointness to the standard discrete elements of pre-Riesz space theory.

What carries the argument

D-disjointness, the broadest of three generalizations of disjointness to partially ordered vector spaces, together with the associated D-discrete elements.

If this is right

D-discrete elements obey several basic order properties in any Archimedean partially ordered space.
D-discrete elements stand in a direct relation to the discrete elements already studied in pre-Riesz spaces.
The equivalence of pervasiveness and weak pervasiveness is guaranteed once finite dimensionality and the Archimedean property are present.

Where Pith is reading between the lines

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

The equivalence may fail without finite dimensionality, pointing to a dimension-sensitive distinction between the two notions.
Similar equivalences could be tested for the other two generalizations of disjointness mentioned in the paper.
The result supplies a concrete simplification when checking order ideals or bands inside low-dimensional ordered spaces.

Load-bearing premise

The space must be both Archimedean and finite-dimensional.

What would settle it

An explicit finite-dimensional Archimedean pre-Riesz space in which a pervasive element fails to be weakly pervasive, or vice versa.

read the original abstract

The notions of disjointness and discrete elements play a prominent role in the classical theory of vector lattices. There are at least three different generalizations of the notion of disjointness to a larger class of partially ordered vector spaces. In recent years, one of these generalizations has been widely studied in the context of pre-Riesz spaces. The notion of $D$-disjointness is the most general of the three disjointness concepts. In this paper we study $D$-disjointness and the related concept of a $D$-discrete element. We establish some basic properties of $D$-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces. We then apply our results to establish the equivalence of pervasiveness and weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces.

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 shows pervasiveness equals weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces by linking them through properties of D-discrete elements.

read the letter

The main thing to know about this paper is that it proves pervasiveness and weak pervasiveness are the same in finite-dimensional Archimedean pre-Riesz spaces. It does so by developing some properties of D-discrete elements and showing that they coincide with discrete elements in the relevant order ideal. What is new is the equivalence itself, presented as a consequence of the earlier results on D-disjointness and D-discrete elements. The paper does a solid job of recalling the three generalizations of disjointness, focusing on the most general D-version, and then deriving the needed facts in Archimedean partially ordered vector spaces. The finite-dimensional assumption lets the argument use direct basis expansions instead of dealing with completion issues, which keeps things straightforward. The soft spots are not major. The equivalence holds only under the stated restrictions, and the paper does not provide much discussion of what happens when those fail, which might have helped readers see the boundaries. Since the full proofs are not in the abstract, one has to trust that the logical sequence holds without gaps, but the stress test suggests the argument is direct. The citation pattern seems appropriate, building on prior definitions without unnecessary self-reference. This is a paper for specialists in the theory of pre-Riesz spaces and ordered vector spaces. Someone working on generalizations of lattice properties or on pervasiveness concepts would get value from the clarification. It is the kind of targeted note that clarifies a distinction in the literature. I would bring this to a reading group if the group is focused on ordered spaces, but otherwise maybe not. It shows serious engagement with the material and deserves peer review to confirm the technical steps.

Referee Report

0 major / 3 minor

Summary. The paper studies D-disjointness and the associated notion of D-discrete elements in partially ordered vector spaces. It derives basic properties of D-discrete elements in the Archimedean setting, relates them to discrete elements in pre-Riesz spaces, and applies these results to prove that pervasiveness and weak pervasiveness are equivalent in every finite-dimensional Archimedean pre-Riesz space.

Significance. If the central equivalence holds, the result simplifies the theory of pre-Riesz spaces in the finite-dimensional case by showing that two pervasiveness notions coincide, thereby removing the need to distinguish them when working with finite bases and order ideals generated by the positive cone. The paper's reduction of the claim to direct verification that every D-discrete element is discrete (and conversely) via finite-dimensional expansions and Archimedean separation of positive and negative parts is a clear strength, as it avoids pathologies associated with infinite suprema or Riesz completions.

minor comments (3)

Abstract: the final sentence states the equivalence but does not name the precise theorem or section where it appears; adding a forward reference would improve readability.
Introduction or preliminary section: the three generalizations of disjointness are mentioned but only D-disjointness is developed in detail; a short explicit comparison of the three notions (even if brief) would help readers situate the contribution.
Notation: ensure uniform use of the D- prefix when referring to D-disjointness and D-discrete elements across definitions, lemmas, and the final equivalence statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript on D-disjointness and D-discrete elements in partially ordered vector spaces, including the recognition of the equivalence between pervasiveness and weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces. We appreciate the recommendation for minor revision and are prepared to incorporate any editorial suggestions. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes equivalence of pervasiveness and weak pervasiveness by showing that D-discrete elements coincide with discrete elements in the order ideal generated by the positive cone. This reduction uses only the Archimedean property (to separate positive and negative parts) and finite dimensionality (to guarantee finite basis expansions), allowing direct verification without infinite suprema or external completions. Prior notions of D-disjointness and pre-Riesz spaces serve as standard background definitions; the central equivalence is proven from these assumptions inside the paper and does not reduce to any fitted parameter, self-definition, or load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard definition of D-disjointness (most general of three variants) and the Archimedean property; no free parameters or new entities are introduced.

axioms (1)

domain assumption The partially ordered vector space is Archimedean
Invoked for the basic properties of D-discrete elements and for the equivalence theorem.

pith-pipeline@v0.9.0 · 5673 in / 1170 out tokens · 30458 ms · 2026-05-18T16:47:48.078762+00:00 · methodology

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/AbsoluteFloorClosure.lean; IndisputableMonolith/Cost/FunctionalEquation.lean reality_from_one_distinction; washburn_uniqueness_aczel unclear

?

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

We establish some basic properties of D-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces. We then apply our results to establish the equivalence of pervasiveness and weak pervasiveness in finite-dimensional Archimedean pre-Riesz spaces.

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

18 extracted references · 18 canonical work pages

[1]

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

work page 2007
[2]

Cristescu, Ordered Vector Spaces and Linear Operators

R. Cristescu, Ordered Vector Spaces and Linear Operators. Abacus Press, England, 1976

work page 1976
[3]

Cristescu, Topological Vector Spaces

R. Cristescu, Topological Vector Spaces. Noordhoff International Publishing, The Netherlands, 1977

work page 1977
[4]

Eriksson, \,J

S.-L. Eriksson, \,J. Jokela\;and\;L. Paunonen, Generalized absolute values, ideals and homomorphisms in mixed lattice groups

work page
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Eriksson-Bique, Generalized Riesz spaces

S.-L. Eriksson-Bique, Generalized Riesz spaces. Rev. Roumaine Math. Pures Appl., No. 36 (1--2), (1991), 47--69

work page 1991
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Eriksson-Bique, Mixed lattice groups

S.-L. Eriksson-Bique, Mixed lattice groups. Rev. Roumaine Math. Pures Appl., No. 44 (2), (1999), 171--180

work page 1999
[7]

van Gaans and A

O. van Gaans and A. Kalauch, Disjointness in partially ordered vector spaces. Positivity 10(3), 573--589 (2006)

work page 2006
[8]

u ck, On Disjointness, Bands and Projections in Partially Ordered Vector Spaces. Positivity and its Applications. Trends in Mathematics. Birkh \

J. Gl \"u ck, On Disjointness, Bands and Projections in Partially Ordered Vector Spaces. Positivity and its Applications. Trends in Mathematics. Birkh \"a user, Cham. (2021). https://doi.org/10.1007/978-3-030-70974-7_7

work page doi:10.1007/978-3-030-70974-7_7 2021
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Jameson, Ordered linear spaces

G. Jameson, Ordered linear spaces. Lecture Notes in Mathematics, vol. 141, Springer-Verlag, Berlin-Heidelberg-New York, 1970

work page 1970
[10]

Jokela, Ideals, bands and direct sum decompositions in mixed lattice vector spaces

J. Jokela, Ideals, bands and direct sum decompositions in mixed lattice vector spaces. arXiv:2202.12651 (2022). https://arxiv.org/abs/2202.12651

work page arXiv 2022
[11]

Jokela, Compatible topologies on mixed lattice vector spaces

J. Jokela, Compatible topologies on mixed lattice vector spaces. arXiv:2204.03459 (2022). https://arxiv.org/abs/2204.03459

work page arXiv 2022
[12]

Jokela, Mixed lattice structures and cone projections

J. Jokela, Mixed lattice structures and cone projections. arXiv:2204.03921 (2022). https://arxiv.org/abs/2204.03921

work page arXiv 2022
[13]

Kalauch\;and\;O

A. Kalauch\;and\;O. van Gaans, Pre-Riesz spaces. Volume 66 of De Gruyter Expositions in Mathematics, De Gruyter, Berlin, 2019

work page 2019
[14]

Kalauch\;and\;H

A. Kalauch\;and\;H. Malinowski, Projection bands and atoms in pervasive pre-Riesz spaces. Positivity, 25: 177--203, 2021. https://doi.org/10.1007/s11117-020-00757-7

work page doi:10.1007/s11117-020-00757-7 2021
[15]

Katsikis and I

V. Katsikis and I. A. Polyrakis. Positive bases in ordered subspaces with the Riesz decomposition property. Studia Math., 174(3): 233--253, 2006

work page 2006
[16]

W. A. J. Luxemburg\;and\;A. C. Zaanen, Riesz Spaces, Volume I. North-Holland Publishing Company, Amsterdam, 1971

work page 1971
[17]

H. H. Schaefer, Banach Lattices and Positive Operators. Springer-Verlag, Berlin, 1974

work page 1974
[18]

B. Z. Vulikh, Introduction to the theory of partially ordered spaces. Wolters--Noordhoff, Groningen, 1967

work page 1967

[1] [1]

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

work page 2007

[2] [2]

Cristescu, Ordered Vector Spaces and Linear Operators

R. Cristescu, Ordered Vector Spaces and Linear Operators. Abacus Press, England, 1976

work page 1976

[3] [3]

Cristescu, Topological Vector Spaces

R. Cristescu, Topological Vector Spaces. Noordhoff International Publishing, The Netherlands, 1977

work page 1977

[4] [4]

Eriksson, \,J

S.-L. Eriksson, \,J. Jokela\;and\;L. Paunonen, Generalized absolute values, ideals and homomorphisms in mixed lattice groups

work page

[5] [5]

Eriksson-Bique, Generalized Riesz spaces

S.-L. Eriksson-Bique, Generalized Riesz spaces. Rev. Roumaine Math. Pures Appl., No. 36 (1--2), (1991), 47--69

work page 1991

[6] [6]

Eriksson-Bique, Mixed lattice groups

S.-L. Eriksson-Bique, Mixed lattice groups. Rev. Roumaine Math. Pures Appl., No. 44 (2), (1999), 171--180

work page 1999

[7] [7]

van Gaans and A

O. van Gaans and A. Kalauch, Disjointness in partially ordered vector spaces. Positivity 10(3), 573--589 (2006)

work page 2006

[8] [8]

u ck, On Disjointness, Bands and Projections in Partially Ordered Vector Spaces. Positivity and its Applications. Trends in Mathematics. Birkh \

J. Gl \"u ck, On Disjointness, Bands and Projections in Partially Ordered Vector Spaces. Positivity and its Applications. Trends in Mathematics. Birkh \"a user, Cham. (2021). https://doi.org/10.1007/978-3-030-70974-7_7

work page doi:10.1007/978-3-030-70974-7_7 2021

[9] [9]

Jameson, Ordered linear spaces

G. Jameson, Ordered linear spaces. Lecture Notes in Mathematics, vol. 141, Springer-Verlag, Berlin-Heidelberg-New York, 1970

work page 1970

[10] [10]

Jokela, Ideals, bands and direct sum decompositions in mixed lattice vector spaces

J. Jokela, Ideals, bands and direct sum decompositions in mixed lattice vector spaces. arXiv:2202.12651 (2022). https://arxiv.org/abs/2202.12651

work page arXiv 2022

[11] [11]

Jokela, Compatible topologies on mixed lattice vector spaces

J. Jokela, Compatible topologies on mixed lattice vector spaces. arXiv:2204.03459 (2022). https://arxiv.org/abs/2204.03459

work page arXiv 2022

[12] [12]

Jokela, Mixed lattice structures and cone projections

J. Jokela, Mixed lattice structures and cone projections. arXiv:2204.03921 (2022). https://arxiv.org/abs/2204.03921

work page arXiv 2022

[13] [13]

Kalauch\;and\;O

A. Kalauch\;and\;O. van Gaans, Pre-Riesz spaces. Volume 66 of De Gruyter Expositions in Mathematics, De Gruyter, Berlin, 2019

work page 2019

[14] [14]

Kalauch\;and\;H

A. Kalauch\;and\;H. Malinowski, Projection bands and atoms in pervasive pre-Riesz spaces. Positivity, 25: 177--203, 2021. https://doi.org/10.1007/s11117-020-00757-7

work page doi:10.1007/s11117-020-00757-7 2021

[15] [15]

Katsikis and I

V. Katsikis and I. A. Polyrakis. Positive bases in ordered subspaces with the Riesz decomposition property. Studia Math., 174(3): 233--253, 2006

work page 2006

[16] [16]

W. A. J. Luxemburg\;and\;A. C. Zaanen, Riesz Spaces, Volume I. North-Holland Publishing Company, Amsterdam, 1971

work page 1971

[17] [17]

H. H. Schaefer, Banach Lattices and Positive Operators. Springer-Verlag, Berlin, 1974

work page 1974

[18] [18]

B. Z. Vulikh, Introduction to the theory of partially ordered spaces. Wolters--Noordhoff, Groningen, 1967

work page 1967