FS-domains are not always RB-domains

Hui Kou; Yuxu Chen; Zhenchao Lyu

arxiv: 2607.00568 · v1 · pith:MFBK4A6Anew · submitted 2026-07-01 · 🧮 math.GN

FS-domains are not always RB-domains

Yuxu Chen , Hui Kou , Zhenchao Lyu This is my paper

Pith reviewed 2026-07-02 01:56 UTC · model grok-4.3

classification 🧮 math.GN

keywords FS-domainsRB-domainsclosed-disk domaindomain theorycounterexampledcporeverse inclusion

0 comments

The pith

Lawson's planar closed-disk domain is an FS-domain but not an RB-domain.

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

The paper proves that the dcpo of all closed disks in the Euclidean plane, ordered by reverse inclusion with the whole plane as bottom element, fails to be an RB-domain. The structure is already established as an FS-domain, so the proof supplies a concrete separation between the two classes. This directly answers a longstanding open question by showing that FS-domains and RB-domains are not identical. The distinction matters for any work that treats the two notions as interchangeable when building approximations or solving domain equations.

Core claim

Lawson's planar closed-disk domain is not an RB-domain. This domain is the dcpo of all closed disks in the Euclidean plane, together with the whole plane as bottom, ordered by reverse inclusion. Since this domain is an FS-domain, it gives a concrete example of an FS-domain which is not an RB-domain, answering negatively the long-standing open problem in domain theory of whether FS-domains and RB-domains are identical.

What carries the argument

The planar closed-disk domain, the dcpo of closed disks under reverse inclusion.

If this is right

FS-domains and RB-domains are distinct classes of dcpos.
The conjecture that every FS-domain is an RB-domain is false.
Theorems that assume FS-domains coincide with RB-domains must be rechecked for this example.
Domain constructions relying on the RB property cannot be applied indiscriminately to all FS-domains.

Where Pith is reading between the lines

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

Similar reverse-inclusion domains over other compact sets may yield additional separations between the two classes.
The topological features of the plane that block the RB property could be isolated to classify which FS-domains remain RB.
Applications that previously used FS and RB interchangeably may need to specify the stronger RB condition when the disk domain appears.

Load-bearing premise

The planar closed-disk domain is an FS-domain.

What would settle it

An explicit construction of an RB-directed set of finite elements whose supremum is the closed unit disk.

read the original abstract

We prove that Lawson's planar closed-disk domain is not an RB-domain. This domain is the dcpo of all closed disks in the Euclidean plane, together with the whole plane as bottom, ordered by reverse inclusion. Since this domain is an FS-domain, it gives a concrete example of an FS-domain which is not an RB-domain, answering negatively the long-standing open problem in domain theory of whether FS-domains and RB-domains are identical.

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 gives a concrete counterexample by proving Lawson's closed-disk domain is not RB, while treating its FS status as given.

read the letter

The main point is that this manuscript proves the closed-disk domain fails to be an RB-domain. Combined with the claim that it is already an FS-domain, the result separates the two classes and answers the open question in the negative.

The new work is the non-RB argument for this specific dcpo: closed disks in the plane under reverse inclusion, with the full plane as bottom element. The construction itself is standard, but showing it lies outside RB-domains supplies the first explicit separation.

The paper does well by working with a geometric, low-dimensional example instead of an abstract existence proof. That choice makes the counterexample easier to inspect and potentially reuse.

The soft spot is the FS direction. The abstract invokes it with "since" and supplies no derivation or citation in the visible text. For the counterexample to stand, the manuscript must either prove the domain is FS or point to a prior rigorous reference. If that step is missing or thin, the separation does not fully go through. The non-RB proof is the load-bearing new part, so its details are what need the closest check.

Citation patterns look ordinary for domain theory. No circularity is visible.

This is for domain theorists who track distinctions among continuous domains. A reader who already knows the definitions of FS and RB will get direct value from seeing the concrete case worked out.

It deserves a serious referee because it targets a stated open problem with a specific, checkable object. A referee can focus on verifying the non-RB argument and the justification for the FS property.

Referee Report

1 major / 0 minor

Summary. The paper proves that Lawson's planar closed-disk domain—the dcpo of all closed disks in the Euclidean plane together with the whole plane as bottom, ordered by reverse inclusion—is not an RB-domain. Since the domain is an FS-domain, the result supplies a concrete counterexample showing that FS-domains are not always RB-domains and thereby answers negatively the open question of whether the two classes coincide.

Significance. If the result holds, the significance is high: it resolves a long-standing open problem in domain theory by exhibiting an explicit geometric counterexample rather than an abstract existence argument. The choice of a familiar, low-dimensional domain makes the separation between FS and RB classes directly verifiable.

major comments (1)

[Abstract] Abstract: the counterexample rests on the assertion that the planar closed-disk domain 'is an FS-domain,' yet the visible text supplies neither a derivation, a lemma, nor a citation for this property. Because the non-RB direction is the claimed contribution, the FS membership is load-bearing; without it the separation claim does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need to substantiate the claim that the planar closed-disk domain is an FS-domain. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the counterexample rests on the assertion that the planar closed-disk domain 'is an FS-domain,' yet the visible text supplies neither a derivation, a lemma, nor a citation for this property. Because the non-RB direction is the claimed contribution, the FS membership is load-bearing; without it the separation claim does not follow.

Authors: We agree that the FS-membership assertion requires explicit support in the text. In the revised version we will add either a short lemma deriving the FS property for this specific domain or a precise citation to the result in the literature (Lawson’s original construction or a standard reference establishing that the closed-disk dcpo is FS). This will make the separation between FS and RB fully self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity detected; non-RB proof is independent of the invoked FS status.

full rationale

The paper's core contribution is a direct proof that the closed-disk domain fails to be an RB-domain. The FS-domain status is invoked via 'since' as a known property of Lawson's construction (external to this manuscript), not derived or fitted inside the paper. No equations reduce to self-definitions, no parameters are fitted then renamed as predictions, and no load-bearing uniqueness theorems are imported via self-citation. The counterexample structure is standard and self-contained once the external FS fact is granted; the new result does not collapse into its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract indicates reliance on standard definitions and properties of dcpos, FS-domains, and RB-domains from domain theory; no free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard definitions and closure properties of FS-domains and RB-domains within the theory of continuous dcpos
The separation result is stated using these established concepts.

pith-pipeline@v0.9.1-grok · 5591 in / 1174 out tokens · 37363 ms · 2026-07-02T01:56:45.727986+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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S. Boyd, L. Vandenberghe,Convex Optimization, Cambridge University Press, Cambridge, 2004

2004

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Abramsky, A

S. Abramsky, A. Jung, Domain theory, in S. Abramsky, D. M. Gabbay, T. S. E. Maibaum (Eds.), Handbook of Logic in Computer Science, Vol. 3, Oxford University Press, 1994, pp. 1–168

1994

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L. C. Evans, R. F. Gariepy,Measure Theory and Fine Properties of Functions, revised ed., CRC Press, Boca Raton, 2015

2015

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G. B. Folland,Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, New York, 1999

1999

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Gierz, K

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott,Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003

2003

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K.Keimel, Bicontinuousdomainsandsomeoldproblemsindomaintheory,Electronic Notes in Theoretical Computer Science257 (2009), 35–54

2009

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Jung,Cartesian Closed Categories of Domains, CWI Tract 66, Centrum voor Wiskunde en Informatica, Amsterdam, 1989

A. Jung,Cartesian Closed Categories of Domains, CWI Tract 66, Centrum voor Wiskunde en Informatica, Amsterdam, 1989

1989

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Jung, The classification of continuous domains, inProceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science (LICS 1990), IEEE Computer Society Press, 1990, pp

A. Jung, The classification of continuous domains, inProceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science (LICS 1990), IEEE Computer Society Press, 1990, pp. 35–40

1990

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A. Jung, R. Tix, The troublesome probabilistic powerdomain,Electronic Notes in Theoretical Computer Science13 (1998), 23 pp

1998

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K.Keimel, Bicontinuousdomainsandsomeoldproblemsindomaintheory,Electronic Notes in Theoretical Computer Science257 (2009) 35–54

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J. D. Lawson, Metric spaces and FS-domains,Theoretical Computer Science405 (2008) 73–74

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R. T. Rockafellar,Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton, 1970. 14

1970