The reviewed record of science sign in
Pith

arxiv: 2605.26906 · v1 · pith:SPYDSUTE · submitted 2026-05-26 · math.GR · math.GN

Polish topologies on endomorphism monoids of linear orders

Serhii Bardyla , Luna Elliott This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-01 15:58 UTCgrok-4.3pith:SPYDSUTErecord.jsonopen to challenge →

classification math.GR math.GN
keywords endomorphism monoidsPolish semigroup topologieslinear ordersautomatic continuityZariski topologypointwise topologysemigroup topologies
0
0 comments X

The pith

The submonoid of infinite-image endomorphisms on the naturals admits only the pointwise Polish semigroup topology, while the full monoids admit infinitely many.

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

The paper introduces property XX, a structural condition on monoids with small groups of units that forces Borel measurable homomorphisms to be continuous. This condition is applied to endomorphism monoids of the orders on the naturals and integers to classify their Polish semigroup topologies. The authors establish that the Zariski topology equals the pointwise topology and is the coarsest Hausdorff semigroup topology on each monoid considered. They prove that End^∞(N,≤) has a unique Polish semigroup topology, while End(N,≤) and End(Z,≤) each have infinitely many distinct ones, and End(N,<) has exactly continuum many with no maximal second-countable topology.

Core claim

The semigroup Zariski topology coincides with the pointwise topology on all monoids considered and is therefore the coarsest Hausdorff semigroup topology. The submonoid End^∞(N,≤) admits a unique Polish semigroup topology, namely the pointwise topology. Despite possessing a finest Polish semigroup topology, the monoids End(N,≤) and End(Z,≤) admit infinitely many distinct Polish semigroup topologies. The monoid End(N,<) admits exactly 2^ℵ₀ Polish semigroup topologies and no maximal second-countable semigroup topology. These conclusions rest on the new property XX, which yields automatic continuity of Borel measurable homomorphisms between the relevant topological semigroups.

What carries the argument

Property XX, the structural condition on monoids that yields automatic continuity of Borel measurable homomorphisms between certain topological semigroups.

If this is right

  • The pointwise topology is the coarsest Hausdorff semigroup topology on each of these monoids.
  • End(N,≤) and End(Z,≤) each possess a finest Polish semigroup topology.
  • End(N,<) admits no maximal second-countable semigroup topology.
  • The semigroup Zariski topology is Polish and coincides with the pointwise topology on all monoids studied.

Where Pith is reading between the lines

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

  • Property XX may classify Polish topologies on endomorphism monoids of additional linear orders such as the rationals.
  • The distinction between unique and multiple topologies appears tied to the presence or absence of finite-image endomorphisms.
  • The methods could extend to other transformation monoids whose unit groups are small.

Load-bearing premise

The monoids End(N,≤), End(Z,≤) and End(N,<) (and the relevant submonoids) satisfy property XX.

What would settle it

A Borel measurable but discontinuous semigroup homomorphism from one of these monoids with a Polish topology into another topological semigroup would contradict the automatic continuity from property XX.

Figures

Figures reproduced from arXiv: 2605.26906 by Luna Elliott, Serhii Bardyla.

Figure 1
Figure 1. Figure 1: Summary of Main Results Proposition 2.3. If X and Y are Polish spaces and f : X → Y is a Borel measurable bijection, then f −1 is Borel measurable also. In particular, if τ1, τ2 are Polish topologies on the same set such that τ1 ⊆ B(τ2), then B(τ1) = B(τ2). Definition 2.4. If A ⊆ X, then we denote the subspace topology that T induces on A by T |A. A subset A of a space X is called meager if A is contained … view at source ↗
read the original abstract

In this paper, we investigate Polish semigroup topologies on the endomorphism monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$. We introduce a new structural condition, property $\mathbb{XX}$, which yields automatic continuity of Borel measurable homomorphisms between certain topological semigroups. This provides a new method for analyzing Polish semigroup topologies on monoids with small groups of units. We show that for all monoids considered, the semigroup Zariski topology coincides with the pointwise topology and is therefore the coarsest Hausdorff semigroup topology. We prove that the submonoid $\operatorname{End}^{\infty}(\mathbb{N},\leq)$ of $\operatorname{End}(\mathbb{N},\leq)$ consisting of all endomorphisms with infinite image admits a unique Polish semigroup topology, namely the pointwise topology. On the other hand, despite possessing a finest Polish semigroup topology, the monoids $\operatorname{End}(\mathbb{N},\leq)$ and $\operatorname{End}(\mathbb{Z},\leq)$, admit infinitely many distinct Polish semigroup topologies. Also, we show that the monoid $\operatorname{End}(\mathbb{N},<)$ admits exactly $2^{\aleph_0}$ Polish semigroup topologies and no maximal second-countable semigroup topology.

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

4 major / 2 minor

Summary. The paper introduces a new structural condition called property XX on monoids with small unit groups, which is used to obtain automatic continuity of Borel measurable homomorphisms. It shows that the semigroup Zariski topology coincides with the pointwise topology (hence the coarsest Hausdorff semigroup topology) on End(N,≤), End(Z,≤), End(N,<) and the submonoid End^∞(N,≤). The central claims are that End^∞(N,≤) admits a unique Polish semigroup topology (the pointwise one), End(N,≤) and End(Z,≤) admit infinitely many distinct Polish semigroup topologies (despite having a finest one), and End(N,<) admits exactly 2^ℵ₀ Polish semigroup topologies with no maximal second-countable semigroup topology.

Significance. If the verifications that the listed monoids satisfy property XX are complete and correct, the work supplies a new method for analyzing Polish semigroup topologies via automatic continuity on monoids with small groups of units. The explicit cardinality statements (unique, infinitely many, continuum-many) would constitute a concrete contribution to the study of topological semigroups arising from linear orders.

major comments (4)
  1. [§2] §2 (definition of property XX and the automatic-continuity theorem): every cardinality claim in §§4–6 rests on the assertion that End(N,≤), End(Z,≤), End(N,<) and End^∞(N,≤) satisfy property XX; the manuscript must supply explicit, self-contained verifications (lemmas or direct checks) that each monoid meets all clauses of the definition, since no independent derivation or machine-checked confirmation is indicated.
  2. [§4] §4 (uniqueness for End^∞(N,≤)): the proof that the pointwise topology is the only Polish semigroup topology invokes automatic continuity from XX; if the verification of XX for this submonoid relies on the infinite-image condition in a way that is not fully spelled out, the uniqueness statement cannot be assessed independently.
  3. [§5] §5 (infinitely many topologies for End(N,≤) and End(Z,≤)): the existence of infinitely many distinct Polish semigroup topologies is derived from the automatic-continuity theorem plus the existence of a finest topology; a concrete pair of distinct topologies whose Polish and semigroup properties are verified without circular appeal to XX would be needed to make the infinitude claim load-bearing.
  4. [§6] §6 (2^ℵ₀ topologies for End(N,<) and absence of maximal second-countable topology): both the exact cardinality and the non-existence of a maximal second-countable semigroup topology are obtained by applying the automatic-continuity result from XX; the manuscript should confirm that the argument does not require any extra hypotheses beyond those already stated for this monoid.
minor comments (2)
  1. [Abstract] The abstract states that the Zariski topology coincides with the pointwise topology for 'all monoids considered'; listing the four monoids explicitly would improve readability.
  2. [Notation section] Notation for the submonoid End^∞(N,≤) is introduced clearly, but ensure that the distinction between ≤ and < is maintained consistently in all statements about the four monoids.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the careful reading and the detailed major comments. We agree that additional explicit verifications will strengthen the manuscript and will revise accordingly. We respond point by point below.

read point-by-point responses

  1. Referee: [§2] §2 (definition of property XX and the automatic-continuity theorem): every cardinality claim in §§4–6 rests on the assertion that End(N,≤), End(Z,≤), End(N,<) and End^∞(N,≤) satisfy property XX; the manuscript must supply explicit, self-contained verifications (lemmas or direct checks) that each monoid meets all clauses of the definition, since no independent derivation or machine-checked confirmation is indicated.

    Authors: We agree that the verifications should be presented as self-contained lemmas. In the revised manuscript we will add four dedicated lemmas (or a single lemma with four parts) in §2 that directly check each clause of property XX for End(N,≤), End(Z,≤), End(N,<) and End^∞(N,≤) using only the internal structure of these monoids. revision: yes

  2. Referee: [§4] §4 (uniqueness for End^∞(N,≤)): the proof that the pointwise topology is the only Polish semigroup topology invokes automatic continuity from XX; if the verification of XX for this submonoid relies on the infinite-image condition in a way that is not fully spelled out, the uniqueness statement cannot be assessed independently.

    Authors: We will expand the verification lemma for End^∞(N,≤) to explicitly isolate the clauses that use the infinite-image condition and show how they are satisfied, thereby making the subsequent appeal to automatic continuity fully independent. revision: yes

  3. Referee: [§5] §5 (infinitely many topologies for End(N,≤) and End(Z,≤)): the existence of infinitely many distinct Polish semigroup topologies is derived from the automatic-continuity theorem plus the existence of a finest topology; a concrete pair of distinct topologies whose Polish and semigroup properties are verified without circular appeal to XX would be needed to make the infinitude claim load-bearing.

    Authors: The finest topology is constructed explicitly before the automatic-continuity theorem is invoked, so the derivation is not circular. Nevertheless, to address the referee’s concern we will add, in the revised §5, an explicit pair of distinct Polish semigroup topologies on each of these monoids together with direct verifications of their Polish and semigroup properties that do not rely on property XX. revision: yes

  4. Referee: [§6] §6 (2^ℵ₀ topologies for End(N,<) and absence of maximal second-countable topology): both the exact cardinality and the non-existence of a maximal second-countable semigroup topology are obtained by applying the automatic-continuity result from XX; the manuscript should confirm that the argument does not require any extra hypotheses beyond those already stated for this monoid.

    Authors: The arguments in §6 use only the verification that End(N,<) satisfies property XX together with the other properties of the monoid already stated in the section. We will insert a short clarifying paragraph at the beginning of §6 that explicitly records this fact. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on independent verification of new property XX

full rationale

The paper introduces property XX as a novel structural condition on monoids with small unit groups, proves that the Zariski topology coincides with the pointwise topology for the monoids considered, verifies that XX holds for End(N,≤), End(Z,≤), End(N,<) and the submonoid End^∞(N,≤), and then applies the resulting automatic-continuity theorem to count Polish semigroup topologies. None of these steps reduces by definition or construction to its own outputs; the verification of XX and the topology-coincidence result are independent mathematical claims. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard axioms of topology and semigroup theory; property XX is a newly defined condition rather than an invented physical entity.

axioms (2)
  • standard math Standard axioms of ZFC set theory and the definition of Polish spaces as separable completely metrizable spaces
    Invoked implicitly when discussing Polish topologies and Borel measurability.
  • domain assumption The pointwise topology and semigroup Zariski topology are well-defined Hausdorff semigroup topologies on the endomorphism monoids
    Used as the baseline topologies whose coincidence is proved.
invented entities (1)
  • property XX no independent evidence
    purpose: Structural condition on topological semigroups that implies automatic continuity of Borel measurable homomorphisms
    Newly introduced in the paper to analyze monoids with small groups of units.

pith-pipeline@v0.9.1-grok · 5750 in / 1523 out tokens · 44493 ms · 2026-07-01T15:58:54.480136+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

27 extracted references · 27 canonical work pages

  1. [1]

    Banakh,Automatic continuity of measurable homomorphisms on Cech-complete topological groups, preprint (2022), arXiv:2206.02481

    T. Banakh,Automatic continuity of measurable homomorphisms on Cech-complete topological groups, preprint (2022), arXiv:2206.02481

    work page arXiv 2022

  2. [2]

    Banakh, I

    T. Banakh, I. Guran, I. Protasov,Algebraically determined topologies on permutation groups, Topology Appl. 159 (2012), 2258–2268

    work page 2012

  3. [3]

    Bardyla, L

    S. Bardyla, L. Elliott, J. Mitchell, Y. P´ eresse,Classifying the Polish semigroup topologies on the sym- metric inverse monoid, Proceedings of the Edinburgh Mathematical Society, published online (2026) 1–30. doi:10.1017/S0013091525101302

    work page doi:10.1017/s0013091525101302 2026

  4. [4]

    T. S. Blyth,On the endomorphism semigroup of an ordered set. Glasgow Mathematical Journal, 37:2 (1995) 173–178

    work page 1995

  5. [5]

    Bodirsky, F

    M. Bodirsky, F. M. Schneider.A topological characterisation of endomorphism monoids of countable structures Algebra Universalis 77:3 (2017), 251–269

    work page 2017

  6. [6]

    Bodirsky, D.M

    M. Bodirsky, D.M. Evans, M. Kompatscher, M. Pinsker.A counterexample to the reconstruction ofω-categorical structures from their endomorphism monoids, Israel Journal of Mathematics 224:1 (2018), 57–82

    work page 2018

  7. [7]

    Coleman,Automorphisms and endomorphisms of first-order structures, PhD Thesis, (2017), https://ueaeprints.uea.ac.uk/id/eprint/64074/

    T.D. Coleman,Automorphisms and endomorphisms of first-order structures, PhD Thesis, (2017), https://ueaeprints.uea.ac.uk/id/eprint/64074/

    work page 2017

  8. [8]

    Elliott,On constructing topology from algebra, PhD Thesis (2022), https://arxiv.org/pdf/2601.13279

    L. Elliott,On constructing topology from algebra, PhD Thesis (2022), https://arxiv.org/pdf/2601.13279

    work page arXiv 2022

  9. [9]

    Elliott, J

    L. Elliott, J. Jonuˇ sas, Z. Mesyan, J. D. Mitchell, M. Morayne and Y. P´ eresse,Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones, Trans. Amer. Math. Soc. 376:11 (2023), 8023–8093

    work page 2023

  10. [10]

    Elliott, J

    L. Elliott, J. Jonuˇ sas, J. D. Mitchell, Y. P´ eresse and M. Pinsker,Polish topologies on endomorphism monoids of relational structures, Advances in Mathematics 431 (2023), 109214

    work page 2023

  11. [11]

    E. D. Gaughan,Topological Group Structures of Infinite Symmetric Groups, Proceedings of the National Academy of Sciences of the United States of America, 58:3 (1967) 907–910

    work page 1967

  12. [12]

    G¨ unt¨ urk¨ un, A

    S. G¨ unt¨ urk¨ un, A. Snowden.The representation theory of the increasing monoid, Memoirs of the American Mathe- matical Society 286 (2023), pp. 148

    work page 2023

  13. [13]

    J. M. Howie,Fundamentals of semigroup theory, Clarendon Press Oxford, London Mathematical Society Mono- graphs,12, 1995

    work page 1995

  14. [14]

    Extending partial isomorphisms of graphs

    Ehud Hrushovski. Extending partial isomorphisms of graphs. Combinatorica, 12(4):411–416, 1992

    work page 1992

  15. [15]

    A. S. Kechris,Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995

    work page 1995

  16. [16]

    Kechris, C

    A. Kechris, C. Rosendal.Turbulence, amalgamation and generic automorphisms of homogeneous structures. Proc. Lond. Math. Soc. 94:2 (2007), 302–350

    work page 2007

  17. [17]

    Kuznetsova,On continuity of measurable group representations and homomorphisms, Studia Math

    Y. Kuznetsova,On continuity of measurable group representations and homomorphisms, Studia Math. 210:3 (2012), 197–208

    work page 2012

  18. [18]

    Marimon, M

    P. Marimon, M. Pinsker.A guide to topological reconstruction on endomorphism monoids and polymorphism clones, preprint (2025) arXiv:2512.01086

    work page arXiv 2025

  19. [19]

    Minimal and in- trinsic topologies on monoids of elementary embeddings

    J. de la Nuez Gonzalez, Z. Ghadernezhad, P. Marimon, M. Pinsker,Minimal and intrinsic topologies on monoids of elementary embeddings, preprint (2026), arXiv:2603.28419v1

    work page arXiv 2026

  20. [20]

    M. Pech, C. Pech,On automatic homeomorphicity for transformation monoids, Monatshefte f¨ ur Mathematik 179:1 (2016), 129–148

    work page 2016

  21. [21]

    Pinsker, C

    M. Pinsker, C. Schindler.On the Zariski topology on endomorphism monoids of omega-categorical structures. The Journal of Symbolic Logic (2023), 1–19

    work page 2023

  22. [22]

    Pinsker, C

    M. Pinsker, C. Schindler (2023).The semigroup of increasing functions on the rational numbers has a unique Polish topology. preprint (2023), arXiv:2305.04921

    work page arXiv 2023

  23. [23]

    Rosendal,Continuity of universally measurable homomorphisms, Forum Math

    C. Rosendal,Continuity of universally measurable homomorphisms, Forum Math. Pi. 7 (2019), e5, 20 pp

    work page 2019

  24. [24]

    Rosendal,Automatic continuity of group homomorphisms, Bull

    C. Rosendal,Automatic continuity of group homomorphisms, Bull. Symbolic Logic 15:2 (2009), 184–214

    work page 2009

  25. [25]

    Rosendal,Automatic continuity in homeomorphism groups of compact 2-manifolds, Israel J

    C. Rosendal,Automatic continuity in homeomorphism groups of compact 2-manifolds, Israel J. Math. 166 (2008), 349–367

    work page 2008

  26. [26]

    Rosendal, L.C

    C. Rosendal, L.C. Suarez,Aspects of automatic continuity, to appear in Bulletin of the American Mathematical Society (2024), arXiv:2406.12143

    work page arXiv 2024

  27. [27]

    Tsankov,Automatic continuity for the unitary group, Proc

    T. Tsankov,Automatic continuity for the unitary group, Proc. Amer. Math. Soc. 141:10 (2013), 3673–3680. S. Bardyla: University of Vienna, Institute of Mathematics, Vienna, Austria. Email address:sbardyla@gmail.com L. Elliott: University of Manchester, Department of Mathematics and Statistics, Oxford Road, Manchester, M13 9PL, UK Email address:luna.elliott...

    work page 2013