A generalization of order convergence

Kazem Haghnejad Azar

REVIEW 2 major objections 6 minor 7 references

A generalization of order convergence

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper defines $F$-order convergence, in which a net in a sublattice converges when a dominating net in a larger ambient lattice decreases to zero, and shows that when the sublattice is $F$-Dedekind complete this agrees with ordinary…

desk verdict A new generalization of order convergence with a solid first half, but the second half's key lemma has a domain error that currently invalidates the b-order continuity results. read the letter →

arxiv 1908.03193 v1 pith:7A2N3XRQ submitted 2019-08-08 math.FA

Kazem Haghnejad Azar This is my paper · ORCID

classification math.FA MSC 47B6546B4046B42

keywords orderconvergenceF-orderb-ordercontinuousoperatorDedekindcompletevectorlatticedoubledualRieszspaceb-property

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Vector lattices carry a classical notion of order convergence: a net converges when the absolute differences are eventually dominated by a decreasing net that reaches zero. This paper broadens the notion by letting the dominating decreasing net live in a larger ambient lattice $F$, and when that ambient lattice is the double order dual $E^{\sim\sim}$ the resulting convergence is called $b$-order convergence. The paper's central claims are that an $F$-Dedekind complete sublattice (one in which every subset bounded above in $F$ has its supremum in $E$) makes $F$-order convergence agree with ordinary order convergence, and that the broader notion supports a coherent theory of $b$-order continuous operators. A sympathetic reader should care because this gives a single framework in which a familiar convergence and a genuinely wider convergence coexist, with precise conditions that pull the wider notion back to the classical one.

What carries the argument

The central object is the $F$-order convergence relation: $x_\alpha \xrightarrow{Fo} x$ iff there is a net $(y_\alpha)$ in $F$ with $y_\alpha \downarrow 0$ in $F$ and $|x_\alpha - x| \le y_\alpha$ for every index. When $F$ is the double order dual $E^{\sim\sim}$ of $E$, the same relation is called $b$-order convergence and written $x_\alpha \xrightarrow{bo} x$. The proof machinery that carries the main arguments is the tail-supremum construction $w_\alpha = \sup_{\beta \ge \alpha} |x_\beta|$: when $E$ is $F$-Dedekind complete, $w_\alpha$ lies in $E$ and forms a decreasing net that dominates $|x_\alpha - x|$, converting an $F$-order null net into an ordinary order null net; in the dual setting $w_\alpha$ is formed in $E^{\sim\sim}$ and is the test object for whether a $b$-order continuous operator preserves $b$-order nullness. Lemma 3.5 applies this construction to characterize positive $b$-order continuous operators.

What would settle it

Take the vector lattice $E = c_0$ inside its double order dual $E^{\sim\sim} = \ell^\infty$. The standard basis net $(e_n)$ is $b$-order convergent to $0$, and the tail supremum $w_n = \sup_{m \ge n} |e_m|$ is the sequence $(0,\ldots,0,1,1,\ldots)$ in $\ell^\infty$, which does not lie in $c_0$. Checking whether a positive operator on $c_0$ that sends every net decreasing $b$-orderly to zero to a $b$-orderly null net nevertheless fails on this non-monotone net $(e_n)$ would settle whether the unstated assumption $w_n \in E$ in Lemma 3.5 is needed.

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Extended reading notes

Core claim

On the paper's own terms: a net $(x_\alpha)$ in a vector sublattice $E$ is $F$-order convergent to $x$ when a net $(y_\alpha)$ in the larger lattice $F$, with the same index set, decreases to zero and dominates $|x_\alpha - x|$; taking $F = E^{\sim\sim}$ gives $b$-order convergence. The paper proves that if $E$ is $F$-Dedekind complete, then every $F$-order convergent net in $E$ is already order convergent in $E$ (Theorem 2.9(1)), and that positive $b$-order continuous operators between Dedekind complete lattices are exactly those operators that send nets decreasing $b$-orderly to zero to nets decreasing $b$-orderly to zero (Lemma 3.5). It also shows that $b$-order continuous operators form a band inside the space of $b$-order bounded operators, and that when both source and target are $b$-Dedekind complete the $b$-order continuous operators coincide with ordinary order continuous operators. The intended upshot is that $b$-order convergence is a controlled broadening of order convergence: it admits a workable operator theory, and structural assumptions such as $F$-Dedekind completeness pull it back to the classical notion.

Load-bearing premise

The proof of Lemma 3.5's converse evaluates $T$ at $w_\alpha = \sup_{\beta \ge \alpha} |x_\beta|$, a supremum taken in the double order dual $E^{\sim\sim}$; the argument silently assumes that each such $w_\alpha$ belongs to $E$, the domain of $T$, and without that assumption the characterization of positive $b$-order continuous operators is not justified.

Editorial extensions

If this is right

In any vector lattice $E$ that is $F$-Dedekind complete, $F$-order convergence and ordinary order convergence produce exactly the same limits, so the broadened convergence adds no new limit points there.
When $E$ is order dense in $F$ and $F$-Dedekind complete, bands in $E$ are exactly the bands in $F$ (Theorem 2.9(2)); with $F = E^{\sim\sim}$ and $E^{\sim} = E^{\sim}_n$, $E$ is perfect (Theorem 2.9(4)).
Between Dedekind complete lattices, a positive operator is $b$-order continuous if and only if it maps every net decreasing $b$-orderly to zero to a net decreasing $b$-orderly to zero, giving a testable criterion (Lemma 3.5 and Corollary 3.6).
If both $E$ and $F$ are $b$-Dedekind complete, the $b$-order continuous operators are precisely the ordinary order continuous operators, so the new class is an extension that differs only outside $b$-Dedekind-complete settings (Proposition 3.7(2)).
The $b$-order continuous operators form a band inside the $b$-order bounded operators, so the structure inherits the usual lattice of ideals and bands (Proposition 3.7(3)).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same $F$-order construction could be applied with $F$ taken to be a Dedekind completion of $E$ rather than the double order dual; the paper does not explore this, but the definitions suggest an ambient-lattice family of convergences indexed by $F$, with classical order convergence at $F = E$ and $b$-order convergence at $F = E^{\sim\sim}$.
Editorial inference: a testable extension would be to check whether the band theorem for $b$-order continuous operators remains true without the Dedekind completeness assumptions in Lemma 3.5, since the proof's tail-supremum construction needs $w_\alpha$ to lie in the domain $E$.
Editorial inference: if the gap in Lemma 3.5 is real, a natural repair is to define $b$-order continuity using only monotone $b$-down-null nets, or to require $E$ to be $b$-Dedekind complete, in which case Theorem 2.9 makes the two definitions coincide.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Here’s my take on Azar’s paper. The new definition—F-order convergence with an ambient lattice F, with b-order convergence as the special case F = E∼∼—is a natural and useful generalization. The basic properties in Theorem 2.4 are correct, the examples (c0 and en) are helpful, and the notion of b-order continuous operators is a sensible analogue of order continuity. The contribution is modest but real: this is not just a repackaging, because the F-order framework is not in the earlier papers on b-property.

The soft spot is exactly where the stress-test lands. In Lemma 3.5, the converse direction takes a b-null net xα, picks yα ∈ E∼∼ with |xα| ≤ yα ↓ 0, and then defines wα = supβ≥α |xβ|. That supremum is computed in E∼∼, and T is only defined on E. Nothing forces wα ∈ E. Dedekind completeness of E does not help: the net is bounded above in E∼∼, not in E. So the expression T wα is unjustified, and the characterization of b-order continuous operators—along with Corollary 3.6 and parts of Proposition 3.7—is not established. This is an internal domain gap, not a disagreement with known results, and it is load-bearing.

There is also a smaller issue: Theorem 2.9(2) has a handwavy line where the proof says “by using Lemma 2.6” after showing B is an ideal in F, but the solidity of B in F is asserted rather than shown. That can probably be fixed; the Lemma 3.5 gap is the real problem.

The first half of the paper is sound and worth having. The second half needs a repaired proof. I’d send this to a referee rather than desk-reject, because the central idea is new and the flaw is a concrete technical repair, not a fundamental conceptual error. The referee should ask for a corrected Lemma 3.5, or a statement with additional hypotheses that make wα land in E.

If the proof is fixed, I’d cite it. As it stands, I wouldn’t rely on the b-order continuity results.

My bottom line: worth engaging with, needs revision.

Referee Report

2 major / 6 minor

Summary. The paper introduces F-order convergence for nets in a sublattice E of a vector lattice F, and specializes it to b-order convergence when F is the double order dual E∼∼. Section 2 collects basic permanence properties (Theorem 2.4), studies F-order closed sets and F-Dedekind completeness, and states results about bands and perfection (Theorem 2.9). Section 3 defines b-order bounded and b-order continuous operators, proves that the latter form a class Ln∼(E,F), and claims a characterization of positive b-order continuous operators in Lemma 3.5 together with consequences in Corollary 3.6 and Proposition 3.7. The central technical result of the second half is Lemma 3.5, but its converse proof contains a domain error: it applies the operator T to a supremum formed in E∼∼ that is not shown to lie in E. As a result, the characterization and the corollary built on it are not established as written.

Significance. If the proof gap were repaired, the paper would offer a useful extension of order convergence and a natural class of operators linked to the existing theory of property (b) and b-order boundedness; the examples distinguishing F-order convergence from ordinary order convergence, such as Example 2.3, are clear and helpful. The paper also has the merit of formulating several structural questions in a concise way. However, the unproved characterization in Lemma 3.5 is load-bearing for the second half of the paper, so the significance of the b-order continuous operator class is currently not fully supported. The contribution is conditional on a successful repair of that argument.

major comments (2)

[Section 3, Lemma 3.5] The converse direction of Lemma 3.5 contains a genuine domain error. From x_alpha bo-converging to 0 the author obtains y_alpha in E∼∼ with |x_alpha| ≤ y_alpha ↓ 0 and then defines w_alpha = sup_{β≥α} |x_β|. This supremum is taken in E∼∼, and the proof immediately applies T to w_alpha. Since T is defined only on E, the expression T w_alpha is unjustified unless w_alpha belongs to E. Dedekind completeness of E does not guarantee this: the set {|x_β| : β ≥ α} is bounded above in E∼∼ but need not be bounded above in E. The inequality 'w_alpha < z_alpha' displayed in the proof is also unexplained. This gap invalidates Lemma 3.5 as stated and propagates to Corollary 3.6, which is proved by invoking Lemma 3.5. A repair would require an additional hypothesis that ensures w_alpha ∈ E (for example a b-property assumption on E) or a different argument avoiding the application of T to elements of E∼∼.
[Section 2, Theorem 2.9(2)] The proof of Theorem 2.9(2) is incomplete. The proof begins 'First we prove that I is an ideal in F', but the symbol I is never defined; the intended object is presumably the band B. More substantively, after showing that B is an ideal in F, the proof asserts 'since E is F-Dedekind complete, by using Lemma 2.6, B is order closed in F'. This does not follow immediately: Lemma 2.6 requires checking that every upward-directed net in B with a supremum in F has that supremum in B. The statement of the theorem may be true, but the argument as written does not supply the required verification.

minor comments (6)

[Abstract and Definition 2.2] The abstract contains the typo 'invistegate' for 'investigate', and both the abstract and Definition 2.2 say 'with the some index set' instead of 'with the same index set'.
[Lemma 2.6] In the first part of the proof, 'Since A is order closed' should read 'Since A is F-order closed'; in the second part, '0 ≤ (|x| − y_α)^+ ↑b |x|' should be '↑F |x|' because the lemma concerns F-order closedness.
[Section 2, after Definition 2.5] The text contains a duplicated phrase 'but but'; it should read 'but the converse in general does not hold'.
[Proposition 3.7] In Proposition 3.7(1), 'sup b T(A) exists in E' should read 'in F'. In the proof of Proposition 3.7(2), the sentence beginning 'Now let T ∈ Ln(E,F)' should refer to Ln∼(E,F), not Ln(E,F).
[Proposition 3.1] The proof of Proposition 3.1 should be expanded: as written, it treats only a net x_α ↑ x'' and does not explicitly verify the definition of b-order boundedness for an arbitrary b-order bounded subset; the converse direction is only asserted. The result is likely true, but the proof should be made fully general.
[Proposition 3.4] The expression '|T|x_α||' in the proof is malformed; it should be written as |T(|x_α|)| = T(|x_α|), since T is assumed positive.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central definitions and theorems are direct, and the only notable defect in Lemma 3.5 is a non-circular domain-error proof gap.

full rationale

The paper's central constructions are definitions, not conclusions extracted from their own statement. F-order convergence (Definition 2.2) is defined via an external majorizing net in F, and b-order convergence is the special case F = E∼∼. Theorem 2.9(1) is proved directly from F-Dedekind completeness by forming suprema of tails inside E; it does not assume the order-convergence conclusion. Sections 2 and 3 use external citations such as [1], [3], and [7] for background facts about b-order boundedness and the b-property; none of these citations is self-citational, and none imports the target theorem. No fitted parameter is renamed as a prediction, no uniqueness claim is imported from the authors' prior work, and no ansatz is smuggled in via self-citation. The proof of Lemma 3.5 does contain a serious gap: in the converse direction, the net w_alpha = sup_{beta >= alpha} |x_beta| is formed in E∼∼, so T w_alpha is undefined unless w_alpha lies in E; Dedekind completeness of E does not force this because w_alpha is only bounded above in E∼∼, not in E. However, this is a domain/deduction error rather than a circular reduction: the hypothesis is not defined in terms of the conclusion, and no equation reduces to its own input by construction. Consequently, Corollary 3.6 and the parts of Proposition 3.7 relying on Lemma 3.5 are not established by the given proof, but that is a correctness risk, not circularity. Overall circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces a definition-based theory and does not fit data. It rests on vector lattice axioms, the standard theory of order duals, and cited results on b-property from [3] and [7]. There are no free parameters or invented entities.

assumptions (3)

domain assumption E is a vector sublattice of a vector lattice F, and F is Dedekind complete in the relevant contexts.
The entire framework of F-order convergence assumes vector lattice structure and the existence of decreasing nets with infimum zero in F. This is stated in the introduction and Definition 2.2.
standard math The canonical embedding Q_E: E -> E sim sim is an order embedding and E sim sim is Dedekind complete.
Used to define b-order convergence and to take suprema in E sim sim in Lemma 3.5; this is standard order dual theory from [1,2].
domain assumption The cited results on b-property and b-order boundedness from [3] and [7] are correct and applicable.
The paper relies on these cited results for Propositions 3.1 and 3.7 and Theorem 2.9; they are not proved in this paper.

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Cite this review

Pith. "Pith review of A generalization of order convergence." pith.science (2026). https://pith.science/paper/7A2N3XRQ

@misc{pith2026190803193,
  author       = {Pith},
  title        = {Pith review of: A generalization of order convergence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7A2N3XRQ}},
  note         = {Machine review of arXiv:1908.03193}
}

abstract

Let $E$ be a sublattice of a vector lattice $F$. $\left( x_\alpha \right)\subseteq E$ is said to be $ F $-order convergent to a vector $ x $ (in symbols $ x_\alpha \xrightarrow{Fo} x $), whenever there exists another net $ \left(y_\alpha\right) $ in $F $ with the some index set satisfying $ y_\alpha\downarrow 0 $ in $F$ and $ | x_\alpha - x | \leq y_\alpha $ for all indexes $ \alpha $. If $F=E^{\sim\sim}$, this convergence is called $b$-order convergence and we write $ x_\alpha \xrightarrow{bo} x$. In this manuscript, first we study some properties of $Fo$-convergence nets and we extend some results to the general case. In the second part, we introduce $b$-order continuous operators and we invistegate some properties of this new concept. An operator $T$ between two vector lattices $E$ and $F$ is said to be $b$-order continuous, if $ x_\alpha \xrightarrow{bo} 0 $ in $E$ implies $ Tx_\alpha \xrightarrow{bo} 0$ in $F$.

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Y. A. Abramovich and C. D. Aliprantis, Locally Solid vector lattices with Appli- cation to Economics , Mathematical Surveys, vol. 105, American Mathematical Society, Providence, RI, 2003

work page 2003
[2]

C. D. Aliprantis and O. Burkinshaw, Positive operators, vol. 119, Springer Sci- ence & Business Media, 2006

work page 2006
[3]

Alpay, B

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work page 2003
[4]

Alpay, B

S ¸. Alpay, B. Altin, and C. Tonyali, A note on vector lattices with property-b , Czechoslovak Mathematical Journal 56 (2006), no. 2, 765–772

work page 2006
[5]

Alpay and B

S ¸ . Alpay and B. Altin, On vector lattices with b-property and strongly order bounded operators, Rendiconti del Circolo Matematico di Palermo 60 (2011), no. 1, 1–12

work page 2011
[6]

Altin, On b-weakly compact operators on Banach lattices , Taiwan

B. Altin, On b-weakly compact operators on Banach lattices , Taiwan. J. Math. 11, (2007), 143-150

work page 2007
[7]

Alpay and Z

S ¸. Alpay and Z. Ercan, Characterizations of Riesz spaces with b-property , Posi- tivity 13 (2009), no. 1, 21–30. Department of Mathematics and Applications, F aculty of Mat he- matical Sciences, University of Mohaghegh Ardabili, Ardab il, Iran. E-mail address : haghnejad@uma.ac.ir

work page 2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Y. A. Abramovich and C. D. Aliprantis, Locally Solid vector lattices with Appli- cation to Economics , Mathematical Surveys, vol. 105, American Mathematical Society, Providence, RI, 2003

work page 2003

[2] [2]

C. D. Aliprantis and O. Burkinshaw, Positive operators, vol. 119, Springer Sci- ence & Business Media, 2006

work page 2006

[3] [3]

Alpay, B

S ¸ . Alpay, B. Altin, and C. Tonyali, On property (b) of vector lattices , Positivity 7 (2003), no. 1, 135–139

work page 2003

[4] [4]

Alpay, B

S ¸. Alpay, B. Altin, and C. Tonyali, A note on vector lattices with property-b , Czechoslovak Mathematical Journal 56 (2006), no. 2, 765–772

work page 2006

[5] [5]

Alpay and B

S ¸ . Alpay and B. Altin, On vector lattices with b-property and strongly order bounded operators, Rendiconti del Circolo Matematico di Palermo 60 (2011), no. 1, 1–12

work page 2011

[6] [6]

Altin, On b-weakly compact operators on Banach lattices , Taiwan

B. Altin, On b-weakly compact operators on Banach lattices , Taiwan. J. Math. 11, (2007), 143-150

work page 2007

[7] [7]

Alpay and Z

S ¸. Alpay and Z. Ercan, Characterizations of Riesz spaces with b-property , Posi- tivity 13 (2009), no. 1, 21–30. Department of Mathematics and Applications, F aculty of Mat he- matical Sciences, University of Mohaghegh Ardabili, Ardab il, Iran. E-mail address : haghnejad@uma.ac.ir

work page 2009