pith. sign in

arxiv: 2503.08691 · v2 · pith:QP2SITN3new · submitted 2025-02-24 · 🧮 math.GN · math.LO· math.RA

On Nondefinability of Interior-Connectedness via the Contact Relation

Rafa{\l} Gruszczy\'nski , Paula Mench\'on This is my paper

Pith reviewed 2026-05-23 02:41 UTC · model grok-4.3

classification 🧮 math.GN math.LOmath.RA
keywords boolean contact algebrasinterior-connectednessnondefinabilitycontact relationregular closed algebrasgeneral topologymereotopology
0
0 comments X

The pith

Interior-connectedness cannot be expressed by means of the contact relation in regular closed algebras.

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 topological property of interior-connectedness cannot be expressed using only the contact relation within regular closed algebras. This nondefinability is established by proving minimality conditions on the algebras and the underlying spaces. A reader would care because it shows the precise limits of the contact relation as an algebraic primitive for capturing topological features. If the result holds, contact alone is insufficient to distinguish interior-connected spaces from those that are not in this algebraic setting.

Core claim

The property of interior-connectedness cannot be expressed by means of contact within regular closed algebras. The authors support this by analyzing the nondefinability and proving certain minimality conditions for algebras and spaces that can be used in demonstrating the result.

What carries the argument

Minimality conditions on algebras and spaces that demonstrate nondefinability of interior-connectedness from the contact relation.

If this is right

  • Regular closed algebras require structure beyond contact to represent interior-connectedness.
  • Similar minimality arguments can establish nondefinability for other topological properties.
  • Contact relations have restricted expressive power for interior topology in algebraic models.

Where Pith is reading between the lines

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

  • Adding an explicit interior operator might be necessary to recover the missing expressivity.
  • The result may apply to other classes of contact algebras beyond the regular closed case.
  • Concrete spaces such as intervals or the real line could serve as test cases for the minimality conditions.

Load-bearing premise

The minimality conditions on algebras and spaces are sufficient to demonstrate that interior-connectedness is not definable from contact.

What would settle it

An explicit first-order formula in the language of contact that defines interior-connectedness exactly on all regular closed algebras would falsify the nondefinability claim.

Figures

Figures reproduced from arXiv: 2503.08691 by Paula Mench\'on, Rafa{\l} Gruszczy\'nski.

Figure 1
Figure 1. Figure 1: The five-point space. In effect, in the regular closed algebras, for any topological space hX, τ i and any regular closed subset A of X we obtain that RC(τ )  c(A) iff A is connected. However, it is natural that the expressive power of the language of BCAs must be topologically limited and that although many topological properties can be captured with the contact relation, there will be such that cannot b… view at source ↗
Figure 2
Figure 2. Figure 2: On the left, we have the frame of open subsets of T with its regular elements marked in cyan; on the right, the Boolean algebra of regular closed subsets of the space. Let us determine the family RO(τ ) of the regular open subsets of T. Firstly, the operation of closure adds only black nodes to red singletons, and so every such singleton is regular open. The closure of {x, z} consumes the two black nodes, … view at source ↗
Figure 3
Figure 3. Figure 3: R is an internally connected element of RC(τ ). x y z a b [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A four-point subspace T \ {a} of the five-point space T. It is obvious that for any set B ∈ RC(τ ), B \ {a} ∈ RO(τ ′ ). Moreover, this transformation does not affect the point b, and so RO(τ ′ ) has the maximal contact relation. Thus the mapping f : RC(τ ) → RC(τ ′ ) such that f(B) := B \ {a} is an isomorphism of BCAs. Observe that for R, f(R) = R\ {a} is not internally connected. Indeed, Intτ ′ f(R) = {x,… view at source ↗
Figure 5
Figure 5. Figure 5: Intτ ′(R \ {a}) is a discrete subspace of T \ {a}, so R \ {a} cannot be internally connected. ∅ {x, a, b} {y, a, b} {z, b} {x, a, y, b} {x, z, a, b} {y, z, a, b} {x, y, z, a, b} ∅ {x, b} {y, b} {z, b} {x, y, b} {x, z, b} {y, z, b} {x, y, z, b} f [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A pictorial presentation of the isomorphism f : RC(τ ) → RC(τ ′ ). The yellow chunks of regions form interiors of the cyan elements of the algebras. 3. The number of elements of the algebra We will show that the proof of Ivanova is the simplest possible in the fol￾lowing sense: Theorem 3.1. If B1 and B2 are isomorphic topological contact algebras with different properties of internal connectedness, then th… view at source ↗
Figure 7
Figure 7. Figure 7: An impossible configuration of atoms on the four-point space S: Atoms {p, q} and {r, s} add up to S. ∅ {p, s} {q, s} {r, s} {p, q, s} {p, r, s} {q, r, s} {p, q, r, s} ∅ {p} {q} {r} {p, q} {p, r} {q, r} {p, q, r} {p, q, r, s} [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The eight-element regular closed algebra with full contact and determined by it the frame of open subsets of S. We can see that S is—up to homeomorphism—the space T \ {a} in [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The regular closed algebra of S with one pair of atoms in contact, {q, s} and {r, s}; and the frame of opens deter￾mined by the algebra. We can see that {q, r, s} is inter￾nally connected clopen set. algebra in which only one pair of atoms is in contact, the only possibility to «spoil» internal connectedness of {q, r, s} is via an automorphism. Since such an automorphism must, in particular, preserve conta… view at source ↗
Figure 10
Figure 10. Figure 10: The regular closed algebra on the four-point set {p, q, r, s} with all atoms separated. Routine verifica￾tion shows that RC(τ ) = RO(τ ) = τ , so this is the algebra of clopen subsets of the space. Acknowledgments This research was funded by the National Science Center (Poland), grant num￾ber 2020/39/B/HS1/00216. We would like to thank Ivo Düntsch for reading various versions of this pa￾per and for stimul… view at source ↗
read the original abstract

This short paper is a small contribution to the field of Boolean contact algebras. We analyze the nondefinability of the property of interior-connectedness, and we prove certain minimality conditions for algebras and spaces that can be used in demonstrating that the aforementioned property cannot be expressed by means of contact within regular closed algebras.

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.

Referee Report

0 major / 2 minor

Summary. The paper establishes minimality conditions on regular closed algebras and the associated topological spaces that suffice to witness the nondefinability of interior-connectedness from the contact relation in Boolean contact algebras, following the standard model-theoretic strategy of producing structures that agree on contact but differ on the target property.

Significance. If the stated minimality conditions are correctly formulated and the accompanying constructions are valid, the result supplies a reusable technical tool for undefinability arguments in contact algebra theory. This is a modest but precise contribution to the literature on the expressive limits of the contact relation.

minor comments (2)
  1. [Abstract] Abstract: the claim that the paper 'proves certain minimality conditions' would be clearer if the abstract briefly indicated the form of those conditions (e.g., 'minimal regular closed algebras on two-point spaces').
  2. The manuscript is short; adding an explicit pair of concrete algebras (or a reference to a figure or example in §2 or §3) that realize the minimality conditions would improve readability without lengthening the paper substantially.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our contribution and the recommendation of minor revision. The report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity; standard model-theoretic undefinability

full rationale

The paper proves minimality conditions on regular closed algebras and spaces to witness that interior-connectedness is not first-order definable from the contact relation. This proceeds by exhibiting pairs of structures that agree on contact yet differ on the target property, which is the canonical model-theoretic method for nondefinability results and does not reduce to any self-definition, fitted input renamed as prediction, or load-bearing self-citation. The derivation chain is self-contained against external model-theoretic benchmarks with no equations or reductions that collapse the result to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, background axioms, or invented entities; full text required to audit the logical setup of regular closed algebras and contact relations.

pith-pipeline@v0.9.0 · 5576 in / 855 out tokens · 21661 ms · 2026-05-23T02:41:53.071799+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    and Düntsch, I

    Bennett, B. and Düntsch, I. (2007). Axioms, algebras and topology. In Aiello, M., Pratt-Hartmann, I., and Van Benthem, J., editor s, Handbook of Spatial Logics, chapter 3, pages 99–159. Springer

    work page 2007

  2. [2]

    Beth, E. (1953). On Padoa’s Method in the Theory of Definit ion. Indaga- tiones Mathematicae (Proceedings), 56:330–339

    work page 1953

  3. [3]

    G., Bennett, B., Gooday, J., and Gotts, N

    Cohn, A. G., Bennett, B., Gooday, J., and Gotts, N. M. (199 7). Qualitative Spatial Representation and Reasoning with the Region Conne ction Calculus. Geoinformatica, 1(3):275–316

    work page

  4. [4]

    and Vakarelov, D

    Dimov, G. and Vakarelov, D. (2006). Contact Algebras and Region-based Theory of Space: A Proximity Approach–I. Fundamenta Informaticae, 74(2– 3):209–249

    work page 2006

  5. [5]

    and Winter, M

    Düntsch, I. and Winter, M. (2005). A representation theo rem for Boolean contact algebras. Theoretical Computer Science , 347(3):498–512

    work page 2005

  6. [6]

    and Grice, M

    Goldblatt, R. and Grice, M. (2016). Mereocompactness an d duality for mereotopological spaces. In Bimbó, K., editor, J. Michael Dunn on Informa- tion Based Logics , pages 313–330. Springer International Publishing, Cham

    work page 2016

  7. [7]

    and Menchón, P

    Gruszczyński, R. and Menchón, P. (2023). From contact re lations to modal operators, and back. Studia Logica, 111(5):717–748

    work page 2023

  8. [8]

    Ivanova, T. (2020). Extended contact algebras and inter nal connectedness. Studia Logica, 108(2):239–254. 12 NON-DEFINABILITY OF INTERNAL CONNECTEDNESS

    work page 2020

  9. [9]

    Johnstone, P. T. (1983). The point of pointless topology . Bulletin (New Series) of the American Mathematical Society , 8(1):41–53

    work page 1983

  10. [10]

    Kontchakov, R., Pratt-Hartmann, I., Wolter, F., and Za kharyaschev, M. (1998). Topology, connectedness, and modal logic. In Krach t, M., de Rijke, M., Wansing, H., and Zakharyaschev, M., editors, Advances in Modal Logic , pages 151–176. CSLI Publications

    work page 1998

  11. [11]

    Kontchakov, R., Pratt-Hartmann, I., Wolter, F., and Za kharyaschev, M. (2010). Spatial logics with connectedness predicates. Logical Methods in Computer Science , 6(3)

    work page 2010

  12. [12]

    and Lemon, O

    Pratt, I. and Lemon, O. (1997). Ontologies for plane, po lygonal mereotopology. Notre Dame J. Formal Logic , 38(2):225–245

    work page 1997

  13. [13]

    and Schoop, D

    Pratt, I. and Schoop, D. (2000). Expressivity in polygo nal, plane mereotopology. Journal of Symbolic Logic , 65(2):822–838

    work page 2000

  14. [14]

    Pratt-Hartmann, I. (2001). Empiricism and rationalis m in region-based theories of space. Fundamenta Informaticae, 46(1–2):159–186

    work page 2001

  15. [15]

    Vakarelov, D. (2007). Region-based theory of space: Al gebras of regions, representation theory, and logics. In Gabbay, D. M., Zakhar yaschev, M., and Goncharov, S. S., editors, Mathematical Problems from Applied Logic II: Logics for the XXIst Century , pages 267–348. Springer, New York, NY. Raf ał Gruszczyński, Orcid: 0000-0002-3379-0577, Paula Menc hón,...

    work page 2007