REVIEW 3 major objections 4 minor 30 references
On the lattice of weak topologies on the bicyclic monoid with adjoined zero
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The lattice of all weak shift-continuous topologies on the bicyclic monoid with adjoined zero is order-isomorphic to the product of two lattices of shift-invariant filters on ω, with the top element 1 adjoined.
desk verdict Bardyla and Gutik classify all weak shift-continuous topologies on the bicyclic monoid with adjoined zero as pairs of shift-invariant filters; the result is likely right and genuinely new, but the proof has two sketched steps that need to be written out. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the shift-invariant filter on $\omega$: a free filter containing every cofinite set and stable under the action of $\mathbb Z$ by translation. The authors read a weak topology $\tau\in{\downarrow}\tau_L$ through the sets $F^U_i=\{n\in\omega:(i,n)\in U\}$ for neighborhoods $U$ of zero, prove the resulting filter is independent of $i$, and show the map $\tau\mapsto\mathcal F_\tau$ is an order isomorphism onto $\mathrm{SIF}_1$; the mirrored construction below $\tau_R$ gives the same result on the second coordinate. Distinct filters are shown to give distinct topologies, and Proposition 1.17 assembles the two one-dimensional descriptions into the product: $\tau_{\mathcal F,\mathcal G}=\tau^L_{\mathcal F}\vee\tau^R_{\mathcal G}$, with $f(\tau_{\mathcal F,\mathcal G})=(\mathcal F,\mathcal G)$ witnessing $\mathcal W\cong \mathrm{SIF}_1\times\mathrm{SIF}_1$.
What would settle it
Exhibit a Hausdorff shift-continuous topology on $C^0$ contained in $\tau_{\min}$ in which some non-zero element $(i,j)$ is not isolated; Lemma 1.9 asserts none exists, and the classification of weak topologies by pairs of filters would fail if one did.
Extended reading notes
Core claim
The paper's central claim is a complete classification: the lattice $\mathcal W$ of all weak shift-continuous Hausdorff topologies on the bicyclic monoid $C^0$ is order-isomorphic to $\mathrm{SIF}_1\times\mathrm{SIF}_1$, where $\mathrm{SIF}_1$ is the lattice of shift-invariant filters on $\omega$ with a top element $1$ adjoined. Here 'weak' means contained in the coarsest inverse semigroup topology $\tau_{\min}$, and 'shift-continuous' means the semigroup operation is separately continuous. Every such topology is uniquely of the form $\tau_{\mathcal F,\mathcal G}=\tau^L_{\mathcal F}\vee\tau^R_{\mathcal G}$: the filter $\mathcal F$ controls one coordinate direction of the neighborhoods of zero, the filter $\mathcal G$ controls the other, and the extreme topologies $\tau_L$ and $\tau_R$ correspond to $\mathcal F=1$ or $\mathcal G=1$. The classification is exact in both directions—every pair of filters gives a valid weak topology and distinct pairs give distinct topologies—so two independent one-dimensional filter lattices carry all the information in the weak-topology lattice.
Load-bearing premise
The entire filter encoding rests on the assumption that every non-zero element of $C^0$ is isolated in any weak shift-continuous topology, which the paper uses at Lemma 1.9 and which is only as sound as the external discreteness theorem for the bicyclic semigroup under semitopological topologies.
Editorial extensions
If this is right
- Every weak shift-continuous topology on $C^0$ takes the form $\tau_{\mathcal F,\mathcal G}=\tau^L_{\mathcal F}\vee\tau^R_{\mathcal G}$ for a unique pair of elements of $\mathrm{SIF}_1$.
- Inclusions between weak topologies are exactly coordinatewise inclusions of the two filters, so the lattice structure of $\mathcal W$ is completely understood once $\mathrm{SIF}_1$ is understood.
- $\mathcal W$ contains an antichain of cardinality $2^{\mathfrak c}$ and a well-ordered chain of cardinality $\mathfrak c$, and $|\mathcal W|=|\mathcal{SCT}|=2^{\mathfrak c}$, so the weak topologies already reach the full size of the shift-continuous lattice.
- There is a well-ordered chain of first-countable weak topologies of order type the tower number $\mathfrak t$, connecting the height of the first-countable part to a set-theoretic cardinal invariant.
- The lattice of weak quasisemigroup topologies is $\mathrm{SIF}_1$: asking for continuous inversion collapses the two independent filters onto their diagonal.
Reading between the lines
- The same two-filter encoding should be attempted for any graph inverse semigroup whose nonzero semigroup part is forced discrete by an external discreteness theorem; the paper's method is stated only for $C^0$, but the machinery is purely filter-theoretic.
- The diagonal collapse $\mathcal W_q\cong\mathrm{SIF}_1$ suggests that stronger continuity requirements on the inversion select retracts of $\mathrm{SIF}_1\times\mathrm{SIF}_1$; one could test intermediate requirements by computing the corresponding subposets.
- Since the weak topologies already have the same cardinality as all shift-continuous topologies, any genuinely new information carried by non-weak topologies must be structural rather than quantitative—a useful constraint when hunting for examples outside $\mathcal W$.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the lattice W of Hausdorff shift-continuous topologies on the bicyclic monoid with adjoined zero C^0 that are contained in the coarsest inverse semigroup topology τ_min. It defines two one-parameter families of topologies τ_L^F and τ_R^G indexed by shift-invariant filters on ω, with an added top element 1 representing τ_L and τ_R, and claims that every weak topology decomposes uniquely as τ_L^F ∨ τ_R^G. On this basis it proves an order isomorphism W ≅ SIF_1 × SIF_1, identifies the sublattice of quasisemigroup topologies with SIF_1, establishes absolute H-closedness of weak topologies, and derives cardinal characteristics: an antichain of size 2^c, well-ordered chains of size c, and first-countable weak topologies forming well-ordered chains of order type t̂, with |W| = |SCT| = 2^c.
Significance. If the main theorem is correct, it reduces the family of weak topologies on C^0 to a product of two independent filter lattices and gives sharp cardinal invariants of that lattice. The construction of τ_L^F and τ_R^G is concrete and checkable, and the cardinal arguments in Section 2 are explicit and reproducible, for example via almost-disjoint families, towers, and free ultrafilters of character c. The paper also gives a clean description of the quasisemigroup sublattice. However, the central classification is currently supported more by assertion than by proof, so the significance is conditional on completing the decomposition argument.
major comments (3)
- [Section 1, Proposition 1.17] The proof does not establish the trichotomy or the uniqueness asserted in the statement. The sentence "It is easy to see that τ satisfies one of the following three conditions" is not a proof that the three conditions exhaust all possibilities for an arbitrary τ ∈ ↓°τ_min; in particular, mixed behavior in individual rows and columns is not ruled out. In each case, the existence of the shift-invariant filter and the claimed base at 0 is asserted by "Similar arguments" without the argument being supplied, and no proof of uniqueness of the pair (τ_1, τ_2) is given. Since Theorem 1.18 uses Proposition 1.17 to define the map f, the classification theorem is not yet supported.
- [Section 1, Theorem 1.18] The phrase "routine verifications" is not enough for an order isomorphism. One must prove that f is well-defined, injective, surjective, order-preserving, and order-reflecting. Surjectivity requires showing that every pair (x,y) ∈ SIF_1 × SIF_1 yields a weak shift-continuous topology τ_{x,y}; well-definedness and injectivity require the uniqueness from Proposition 1.17; order-reflection requires showing that τ_{x,y} ≤ τ_{x',y'} implies x ≤ x' and y ≤ y'. None of these steps is written out, and the theorem is load-bearing for the central claim W ≅ SIF_1 × SIF_1.
- [Section 1, Lemma 1.9] The assertion "Since each non-zero point is isolated in (C^0, τ)" is used to prove that the families F_i are closed under supersets, but no proof or reference is given at that point. The fact follows from the Bertman–West theorem [14] applied to the subsemigroup C with the restricted topology, but this needs to be stated explicitly, because the entire filter encoding in Lemmas 1.9–1.13 depends on it.
minor comments (4)
- [Throughout] There are several typographical errors, including "we ak" in the abstract, "T he" and "sh ift" in the first pages; an editorial pass is needed.
- [Section 2, Theorem 2.2(1)] The expression "[w]^ω" should be "[ω]^ω".
- [Section 1, SIF_1 definition] The symbol ⊂ is used both for strict and non-strict inclusion; in the definition "F ≤ G iff G = 1 or F ⊂ G", the intended meaning should be stated explicitly (reflexivity requires F ≤ F to hold).
- [Section 1, Proposition 1.17] The notations τ_{F,1}, τ_{1,G}, τ_{F,G}, and τ_{1,1} are introduced before the correspondence with SIF_1 is defined; it would help to summarize all these cases in one displayed definition.
Circularity Check
No circularity: the main theorem is a constructed bijection between topologies and shift-invariant filters, with no fitted parameter or self-citation chain doing the work.
full rationale
The derivation chain is self-contained in the relevant sense. Lemmas 1.9 and 1.11 associate to every topology in the lower cone of τ_L a shift-invariant filter, and Lemma 1.12 constructs a topology from each such filter; Theorem 1.13 proves the order isomorphism, and the right-cone analogue is Theorem 1.16. Proposition 1.17 then decomposes W into ↓τ_L × ↓τ_R. The map is not defined by the target isomorphism: the filters are extracted from neighborhood bases and the topologies are built from the filters independently. No parameter is fitted to a subset of data and renamed a prediction; the cardinal-characteristic results are derived from external set-theoretic facts about filters on ω. The cited results [7, Theorem 3.6] and [23, Theorem 1]/[8, Lemma 3] are self-citations of the authors, but they are independent supporting theorems about the existence of τ_min and the least shift-continuous topology, and neither presupposes the classification of weak topologies being proved here. The main proof gap is the terseness of Proposition 1.17 ('It is easy to see' and 'routine verifications' in Theorem 1.18), but a missing proof of uniqueness/decomposition is a rigor concern, not a circular reduction: the claim does not assume its conclusion. Thus no circular step can be exhibited with a specific equation or fitted parameter.
Assumptions & free parameters
assumptions (5)
- standard math ZFC set theory, including existence of ultrafilters and almost disjoint families of size c
- domain assumption The bicyclic semigroup C admits only the discrete semitopological semigroup topology [14]
- domain assumption The coarsest inverse semigroup topology tau_min on C^0 has open neighborhood base B_min(0) = {C_n : n in omega} [7, Theorem 3.6]
- domain assumption Every congruence on the bicyclic monoid is a group congruence and C/sigma is isomorphic to the additive group of integers [28, Theorem 3.4.5]
- standard math There exists a free ultrafilter on omega of character c [29, Theorem 4.4.2]
Cite this review
Pith. "Pith review of On the lattice of weak topologies on the bicyclic monoid with adjoined zero." pith.science (2026). https://pith.science/paper/4KQJURCR
@misc{pith2026190804566,
author = {Pith},
title = {Pith review of: On the lattice of weak topologies on the bicyclic monoid with adjoined zero},
year = {2026},
howpublished = {\url{https://pith.science/paper/4KQJURCR}},
note = {Machine review of arXiv:1908.04566}
}
abstract
A Hausdorff topology $\tau$ on the bicyclic monoid with adjoined zero $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice of all shift-invariant filters on $\omega$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.
Reference graph
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