{"id":"54c95c72-1ffd-43a5-8a51-3fb78dd7fb27","arxiv_id":"1908.04566","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The lattice of weak shift-continuous topologies on C^0 is isomorphic to the product of two copies of the poset of shift-invariant filters with a top element attached, and it contains antichains of size 2^c and chains of size c and t.","lead":"This paper classifies all weak shift-continuous topologies on the bicyclic monoid with adjoined zero: each is determined by two shift-invariant filters on the natural numbers, one for each side of the semigroup. The classification makes the whole lattice measurable, yielding antichains and chains of the largest possible cardinalities and connecting the lattice to the tower number t.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unproved decomposition in Proposition 1.17 is the true load-bearing step; Theorem 1.18's 'routine' order-isomorphism is not established until the trichotomy and uniqueness are written out.","rationale":"I read the paper in good faith and found the main construction plausible. The external theorem cited by the reader is actually a well-matched dependency: Bertman and West [14] treat semitopological semigroups, and Hausdorffness ensures the non-zero part C is open in C0, so Lemma 1.9's isolation step is supported. The most load-bearing weakness is instead the skeletal proof of Proposition 1.17, which underpins the entire isomorphism: the trichotomy is asserted without derivation and uniqueness is not addressed. Theorem 1.18 merely calls the order-isomorphism verification routine. Since these two steps are exactly the central claim, a conditional verdict is appropriate until the decomposition is fully proved. Separately, the cardinality bound in Lemma 2.4 appears to have a typo or gap: a filter generated by a countable base plus one coinfinite set can already have cardinality c, so the assertion |G_δ| < c needs a base-sized argument; this affects the well-ordered chain result in Theorem 2.5(2) but not the central lattice classification.","tokens_in":14134,"tokens_out":34803,"duration_ms":344185,"concrete_test":"Write out a complete proof of Proposition 1.17. For τ in the lower cone of τ_min, fix n with U \\ C_n infinite for every neighborhood U of 0, and define A_U to be the set of rays among the first n rows and columns that meet U infinitely. Prove that the family (A_U) is closed under finite intersections, hence has a common ray; then use shift-continuity to propagate that ray and force one of the three claimed cases. Verify uniqueness by showing the row-trace filter and the column-trace filter are recovered from τ alone. If this derivation cannot be completed, or if a concrete topology with disjoint A_U and A_V is found, the classification theorem is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim reduces W to SIF_1 x SIF_1 through Proposition 1.17 and Theorem 1.18. Proposition 1.17 asserts that every τ in the lower cone of τ_min satisfies one of three conditions and has a unique decomposition τ = τ1 ∨ τ2 with τ1 in ↓τL and τ2 in ↓τR. The proof, however, only sketches the trichotomy with 'It is easy to see' and never proves uniqueness; Theorem 1.18 then says 'routine verifications' give the order isomorphism. If the trichotomy has a missing case, or if a topology admits two different filter pairs, the claimed bijection W ≅ SIF_1 x SIF_1 fails. The reader's cited concern about Lemma 1.9 is less acute: reference [14] is explicitly about semitopological semigroups, and Hausdorffness makes C = C0 \\ {0} open, so non-zero points are isolated from that theorem. The real unverified core is the decomposition and injectivity in Proposition 1.17.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14278,"tokens_out":18809,"duration_ms":202639,"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":[{"comment":"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":"Section 1, Proposition 1.17"},{"comment":"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":"Section 1, Theorem 1.18"},{"comment":"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.","section":"Section 1, Lemma 1.9"}],"minor_comments":[{"comment":"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":"Throughout"},{"comment":"The expression \"[w]^ω\" should be \"[ω]^ω\".","section":"Section 2, Theorem 2.2(1)"},{"comment":"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":"Section 1, SIF_1 definition"},{"comment":"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.","section":"Section 1, Proposition 1.17"}],"recommendation":"major_revision","confidential_remarks":"The result appears likely to be correct, and the paper fits the journal's scope. However, the proof of the central decomposition in Proposition 1.17 and the order-isomorphism claim in Theorem 1.18 are sketches rather than proofs. I would ask the editor to require a complete proof of these two results before publication, and also to verify that reference [14] is being applied correctly to semitopological (rather than only topological) semigroups, since Lemma 1.9 relies on that fact without citing it at the point of use."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: Bardyla and Gutik reduce the lattice W of all weak shift-continuous topologies on C^0 to SIF_1 x SIF_1, with each topology of the form tau_L^F ∨ tau_R^G for a unique pair of shift-invariant filters F, G. That is a complete structural description of a lattice where previously only the extremes tau_L, tau_R and the coarsest inverse semigroup topology tau_min were understood. The cardinal corollaries — an antichain of size 2^c, a well-ordered chain of size c, and a first-countable chain of order type t — are nice and follow from standard filter constructions. No fitting, no invented entities, no circularity. The paper builds on prior results, including the authors' own, in an honest way.\n\nWhat the paper does well: the topology constructions tau_L^F and tau_R^G are concrete, Lemmas 1.9–1.12 give an explicit filter encoding, and the later cardinal arguments are mostly checkable. The citation pattern looks fair; the paper cites independent prior theorems, including Bertman–West, and does not lean on itself to a suspicious degree.\n\nThe soft spots are in the middle. Proposition 1.17 asserts a trichotomy with “it is easy to see” and never proves uniqueness of the decomposition tau = tau_1 ∨ tau_2. Theorem 1.18 then claims the order isomorphism after “routine verifications.” Those are load-bearing steps: if a topology admitted two different filter pairs, or if the trichotomy missed a case, W would not be isomorphic to SIF_1 x SIF_1. I think the result is true and provable by writing out the cases, but as written the proof is incomplete. The reader’s concern about Lemma 1.9 is less acute: reference [14] is explicitly about semitopological semigroups, and Hausdorffness makes C open, so the claim that non-zero points are isolated is grounded — but the paper should say so at that point. Lemma 2.3(4) and Lemma 2.4 are also asserted rather than demonstrated, but those are secondary and I believe them.\n\nBottom line: this is a genuine contribution, and it deserves a serious referee. The right verdict is major revision, with the decomposition and order-isomorphism proofs expanded. I would not desk reject it; I would send it out.","headline":"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.","tokens_in":14853,"tokens_out":2897,"would_cite":true,"duration_ms":28406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A15","06B23"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["lattice of topologies","bicyclic monoid","weak topology","shift-continuous topology","shift-invariant filter","inverse semigroup","cardinal characteristics","tower number"],"falsifier":"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.","tokens_in":13887,"feed_emoji":"🔀","tokens_out":14055,"duration_ms":123823,"temperature":0.7,"pith_summary":"The bicyclic monoid with zero adjoined is an infinite semigroup with many Hausdorff topologies; the weak ones are those contained in the coarsest inverse semigroup topology, with multiplication separately continuous. This paper proves a complete classification: every weak shift-continuous Hausdorff topology on $C^0$ is $\\tau_{\\mathcal F,\\mathcal G}=\\tau^L_{\\mathcal F}\\vee\\tau^R_{\\mathcal G}$ for a unique pair of shift-invariant filters on $\\omega$, and the whole lattice $\\mathcal W$ is order-isomorphic to $\\mathrm{SIF}_1\\times\\mathrm{SIF}_1$. A filter on the left coordinate and a filter on the right coordinate, together, contain all the information about how neighborhoods of zero look; nothing else is needed. The paper then derives cardinal geometry: $\\mathcal W$ has antichains of size $2^{\\mathfrak c}$, well-ordered chains of length $\\mathfrak c$, and first-countable chains of order type the tower number $\\mathfrak t$, and $|\\mathcal W|=|\\mathcal{SCT}|=2^{\\mathfrak c}$. The result matters because it converts a potentially enormous topological lattice into a one-dimensional filter lattice that can be studied with set-theoretic tools.","feed_headline":"Every weak topology on the bicyclic monoid is two filters","feed_subtitle":"The lattice reduces to pairs of shift-invariant filters on ω, with antichains of size 2^c and chains of length c.","key_machinery":"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 $\\mathcal W\\cong \\mathrm{SIF}_1\\times\\mathrm{SIF}_1$.","core_discovery":"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.","pith_inferences":["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 $\\mathcal W$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the discreteness theorem for the bicyclic semigroup under semitopological topologies; the filter encoding assumes all non-zero points are isolated.","marker":"[14]"},{"why":"Defines the coarsest inverse semigroup topology $\\tau_{\\min}$ on $C^0$ and its neighborhood base, the envelope relative to which 'weak' is defined.","marker":"[7]"},{"why":"Provides the unique compact shift-continuous topology $\\tau_c$ and shows it is the infimum of $\\mathcal{SCT}$, which makes the shift-continuous poset a complete lattice.","marker":"[23]"},{"why":"Gives the existence of an almost disjoint family of cardinality $\\mathfrak c$, used to build the antichain of first-countable weak topologies.","marker":"[27]"},{"why":"Supplies an ultrafilter on $\\omega$ of character $\\mathfrak c$, used to construct the well-ordered chain of free filters of cardinality $\\mathfrak c$.","marker":"[29]"},{"why":"Defines the tower number $\\mathfrak t$ and the bound $\\omega_1\\le\\mathfrak t\\le\\mathfrak c$, used for the first-countable chains of order type $\\mathfrak t$.","marker":"[20]"}],"fun_headline_variants":["Weak topologies on bicyclic monoid are filter pairs","Bicyclic monoid weak topology lattice from two filters","Antichains of 2^c in weak topology lattice","Filter pairs classify weak topologies on C^0"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Weak topologies on bicyclic monoid are filter pairs","Bicyclic monoid weak topology lattice from two filters","Antichains of 2^c in weak topology lattice","Filter pairs classify weak topologies on C^0"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000595,"raw_usage":{"total_tokens":2803,"prompt_tokens":978,"completion_tokens":1825,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":1758}},"tokens_in":594,"tokens_out":1825,"duration_ms":13615,"temperature":1.0,"reasoning_tokens":1758,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:40:38.772508+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bertman, T","cited_arxiv_id":null,"evidence_quote":"Supplies the discreteness theorem for the bicyclic semigroup under semitopological topologies; the filter encoding assumes all non-zero points are isolated."},{"cited_title":"Bardyla, On universal objects in the class of graph inverse semigroup s, European Journal of Mathematics, to appear, DOI: 10.1007/s40879-018-0300-7","cited_arxiv_id":null,"evidence_quote":"Defines the coarsest inverse semigroup topology $\\tau_{\\min}$ on $C^0$ and its neighborhood base, the envelope relative to which 'weak' is defined."},{"cited_title":"Kunen Set theory , Studies in Logic and the Foundations of Mathematics, vol","cited_arxiv_id":null,"evidence_quote":"Gives the existence of an almost disjoint family of cardinality $\\mathfrak c$, used to build the antichain of first-countable weak topologies."},{"cited_title":"van Mill, An introduction to β (ω ), in: Handbook of set-theoretic topology, North-Holland, Amste rdam, (1984), 503–568","cited_arxiv_id":null,"evidence_quote":"Supplies an ultrafilter on $\\omega$ of character $\\mathfrak c$, used to construct the well-ordered chain of free filters of cardinality $\\mathfrak c$."},{"cited_title":"van Douwen, The integers and topology , in: Handbook of set-theoretic topology, North-Holland, Amster dam, (1984), 111–167","cited_arxiv_id":null,"evidence_quote":"Defines the tower number $\\mathfrak t$ and the bound $\\omega_1\\le\\mathfrak t\\le\\mathfrak c$, used for the first-countable chains of order type $\\mathfrak t$."}],"review_version":1}