{"id":"bb3ead96-b1f3-4936-99a3-0ae0606fea12","arxiv_id":"1908.06236","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Sorgenfrey line admits no T1-complementary Hausdorff paratopological group topology, and new PT-sequence criteria characterize transversality for paratopological groups.","lead":"This paper proves that the Sorgenfrey line, a standard topological space built from half-open intervals, has no Hausdorff paratopological group topology that combines with it into a discrete topology and intersects it only in cofinite sets. It also introduces PT-sequences and gives criteria for transversality in paratopological groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified to the Sorgenfrey-line claim; Theorem 3.3's key product-open step is valid.","rationale":"I focused on the reader's strongest claim, the negative answer for the Sorgenfrey line. A careful check of Theorem 3.3 shows that its most suspicious-looking inference is actually valid: the product UO of a σ-open neighborhood and a τ-open neighborhood is open in both topologies, so T1-independence forces its complement to be finite. This makes the density argument and the final contradiction with transversality rigorous. The reader's weakest-assumption concern about Shakhmatov's theorem is real for Theorem 3.6, but that theorem is not needed for the Sorgenfrey-line claim. The paper still contains compressed proofs and several external dependencies, so I do not argue for a stronger verdict than the reader's conditional one; I simply find no load-bearing flaw in the central claim.","tokens_in":19044,"tokens_out":42462,"duration_ms":440068,"concrete_test":"Recompute the key step in Theorem 3.3 by explicitly checking that UO, for U∈σ(e) and O∈τ(e), is open in both σ and τ via right-translation and left-translation homeomorphisms; if confirmed, the proof of Corollary 3.4 is complete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 3.3 with Corollary 3.4, is supported by a sound argument. The potentially dangerous step is the inference that G\\UO is finite from T1-independence. That step is justified: U∈σ(e) and O∈τ(e) imply UO is open in both σ and τ, because UO is the union of the τ-open left translates uO for u∈U, and also the union of the σ-open right translates Uo for o∈O, left and right translations being homeomorphisms in a paratopological group. Since σ and τ are T1-independent, every nonempty set open in both is cofinite; hence G\\UO is finite. The subsequent density claim, the infinite-intersection claim, and the final contradiction with transversality all follow. I could not identify a load-bearing gap in the central Sorgenfrey-line result; the external-theorem concerns raised by the reader affect Theorem 3.6 and the extension results, not Theorem 3.3 itself.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies transversality, T1-independence, and T1-complementarity for paratopological group topologies. Its main headline result is Theorem 3.3 and Corollary 3.4: no infinite Hausdorff SD-paratopological group, and in particular the Sorgenfrey line, admits a T1-complementary Hausdorff paratopological group topology, giving a negative answer to Problem 3.2 (Problem 10 of [5]). The paper also introduces submaximal paratopological group topologies (Theorems 4.2 and 4.6), an extension theorem for central subgroups (Theorem 4.11), and develops PT-sequences and PT-filters, proving a characterization of Abelian paratopological groups determined by PT-sequences (Theorem 5.9), a sequentiality/non-Fréchet-Urysohn result (Theorems 5.14 and 5.15), the nonexistence of T1-complementary topologies for the strongest topology determined by a T-sequence (Theorem 5.20), and several results on countable groups and PT-filters in Section 6.","tokens_in":19221,"tokens_out":20938,"duration_ms":182526,"significance":"If the main theorem is correct, it resolves Problem 10 of [5] in the negative, a result that is likely to be of interest to researchers working on paratopological groups and lattice-theoretic complementarity. I have checked the proof of Theorem 3.3 in detail and found it sound: the claim that UO is open in both σ and τ is valid because left and right translations are homeomorphisms in a paratopological group, the use of T1-independence to conclude that G\\UO is finite is legitimate, and the final density/transversality contradiction is correct. The paper also offers useful new tools, especially the submaximal paratopological group topology and PT-sequences. However, several other advertised results, specifically Theorems 3.5, 3.6, 4.9, and 5.20, contain proof gaps or misquotations that must be addressed before the paper can be accepted.","major_comments":[{"comment":"The proof of Theorem 4.9 is incomplete. In the case where N = ker f is neither τ-open nor τ-discrete, the assertion that \"it is obvious that one can get a finer non-discrete paratopological group topology on G by taking N open\" is not justified: one must check that the topology generated by adding N as a neighborhood of e is a paratopological group topology, and that it is not discrete. In the discrete case, the family A is asserted to be a filter base determining a non-discrete paratopological group topology η with η > τ, but the axioms for a neighborhood base of a paratopological group topology are not verified, and it is not shown that η is strictly finer than τ or that η is non-discrete. Because Theorem 4.9 is used in the proof of Theorem 4.11 and Corollary 4.12, these omissions affect the central-subgroup extension result advertised in the abstract.","section":"Section 4 (Theorem 4.9)"},{"comment":"Theorem 3.5 is stated with \"it easily prove\" after the proof of Theorem 3.3, but the argument of Theorem 3.3 does not adapt directly. In the complementary (rather than T1-complementary) setting, a nonempty set UO open in both σ and τ must be equal to G because the intersection of the two topologies is indiscrete, so the contradiction G\\UO finite used in Theorem 3.3 is not available. The authors should provide a complete proof of Theorem 3.5 or remove the claim.","section":"Section 3 (Theorem 3.5)"},{"comment":"In the proof of Theorem 3.6, the passage from a Hausdorff paratopological group topology τ_i to a coarser Hausdorff paratopological group topology σ_i of countable weight relies on Shakhmatov's result [13] and [2, Problem 5.2.A]. Since [2] is mainly about topological groups, it is not clear that the quoted statement applies verbatim to paratopological groups, where inversion need not be continuous. Please state the exact result and explain why it holds for paratopological groups; if it does not, the equivalence in Theorem 3.6 is unproved, although Theorem 3.3 is unaffected.","section":"Section 3 (Theorem 3.6)"},{"comment":"The statement of Proposition 5.19 has the topologies swapped relative to its use in Theorem 5.20. As printed, the proposition says that if σ is sequential then the space (G,σ) is countably compact and has no nontrivial convergent sequences; in Theorem 5.20 it is τ = P(G|{a_n}) that is sequential, so the printed proposition does not justify the conclusion that (G,σ) is countably compact. Please correct the statement of Proposition 5.19 or the proof of Theorem 5.20, and verify the quotation from [15, Proposition 2.4].","section":"Section 5 (Proposition 5.19 and Theorem 5.20)"}],"minor_comments":[{"comment":"There are typographical errors such as \"adimit\" (Abstract and Corollary 3.4), \"transveral\" (Abstract), and \"B/suppress laszczyk\" (Introduction); these should be corrected.","section":"Throughout"},{"comment":"In the final transversal step of the proof of Theorem 3.3, the statement should read \"there exist U∈σ(e) and V∈τ(e) such that U∩V={e}\", not \"there exist U∈σ(e) and V∈σ(e)\", and the identity element should be denoted consistently as e rather than 0.","section":"Section 3 (Theorem 3.3 proof)"},{"comment":"In Theorem 4.6 and its proof, the expression \"τ /notlessorslnteql PM G\" is garbled; it should be \"τ ≰ PM_G\".","section":"Section 4 (Theorem 4.6)"},{"comment":"In the proof of Theorem 5.14, \"0 ∈ F \\ F\" should read \"0 ∈ \\overline{F} \\ F\".","section":"Section 5 (Theorem 5.14 proof)"},{"comment":"The notation P(G|{a_n}) is used with the index set ω in most of Section 5, but Theorem 5.20 uses {a_n}_{n∈N}; please standardize the indexing throughout.","section":"Section 5 (notational consistency)"},{"comment":"In Theorem 6.12, \"there exits\" should be \"there exists\", and in Definition 6.3 the notation \"{Dn}nω\" should be \"{D_n}_{n∈ω}\".","section":"Section 6 (Theorem 6.12, Definition 6.3)"},{"comment":"The abstract says \"a non-discrete paratopological group topology G contains a central subgroup\", but the corresponding theorem is about a group H containing a subgroup G; please rephrase to avoid ambiguity.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The Sorgenfrey-line result is sound and appears to be the paper's main publishable contribution. The major concerns are concentrated in auxiliary theorems: Theorem 4.9 needs a complete proof, Theorem 3.5 needs either a proof or removal, Theorem 3.6 needs verification that the cited externality applies to paratopological groups, and Proposition 5.19 needs a corrected statement. These are substantial but appear fixable without changing the central result, so I recommend a major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main event is Theorem 3.3 and its Corollary 3.4: the Sorgenfrey line does not admit a T1-complementary Hausdorff paratopological group topology. That answers Problem 10 of the 2017 survey, and I think the proof is correct. The tricky step, where T1-independence forces G\\UO finite because UO is open in both topologies, works exactly as the stress-test note says. I checked the product-open logic and it holds. So the central claim is solid.\n\nThe paper also gives a nice criterion for transversality in terms of the submaximal paratopological group topology PM_G, proves that PM_G is always a topological group, and introduces PT-sequences and PT-filters as tools. Those are genuinely reusable, and they transplant the Protasov–Zelenyuk machinery into paratopological groups in a plausible way. Theorem 5.9, the characterization of PT-sequences, is a reasonable analogue of the known T-sequence criterion. Credit where due: these are useful working tools, not just a one-off counterexample.\n\nWhere it gets softer is the extension material. The proof of Theorem 4.9 is more of a sketch than a proof. The filter base A is asserted to determine a finer non-discrete topology, but it is not shown to be a neighborhood base, and the maximality argument is compressed. Same for Theorem 5.20: the jump from “σ is ω-narrow” to “σ is saturated” relies on the Alas–Sanchis results and is stated too fast. These may well be repairable, but they need real checking.\n\nTwo more concerns, both moderate. Theorem 3.6 leans on Shakhmatov’s theorem that every Hausdorff paratopological group topology on a countable group has a coarser Hausdorff paratopological group topology with countable weight. I believe that is in the literature, but since the proof is not included, the equivalence between T1-independence and transversal T1-independence on countable groups stands on that external result. The other issue is the typographical state: “adimit,” “B/suppress laszczyk,” “τ /notlessorslnteqlPM G” and similar corruption appear throughout. It is readable, but it badly needs a proofreading pass.\n\nBottom line: the central result is correct and the PT-sequence framework is a genuine addition to the paratopological toolbox. The paper should go to a serious referee, with the expectation that the sketched proofs in Sections 4 and 5 get tightened before publication. Specialists in paratopological groups and the lattice of topologies will want to read this. It is not going to reorganize any major branch, but it resolves a posed open problem and gives useful criteria.","headline":"The Sorgenfrey-line negative answer to Problem 10 is real and worth publishing; the paper also contributes reusable devices (SD-paratopological groups, PT-sequences) for the paratopological group topology lattice, but some extension proofs are sketches and the text needs a cleanup pass.","tokens_in":19771,"tokens_out":1137,"would_cite":true,"duration_ms":13492,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22A05","54H11","54A25","54A35","54G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any infinite Hausdorff SD-paratopological group—and in particular the Sorgenfrey line—no Hausdorff paratopological group topology is T1-complementary.","keywords":["paratopological groups","transversal topologies","T1-independent topologies","T1-complementary topologies","Sorgenfrey line","PT-sequence","PT-filter","submaximal paratopological group topology"],"falsifier":"Exhibit a Hausdorff paratopological group topology $\\sigma$ on $\\mathbb{R}$ such that, with $\\tau$ the Sorgenfrey topology, $\\tau\\cap\\sigma$ is the cofinite topology and $\\tau\\vee\\sigma$ is the discrete topology; such a pair would refute Corollary 3.4 and, with it, Theorem 3.3.","tokens_in":18806,"feed_emoji":"🚫","tokens_out":21137,"duration_ms":171472,"temperature":0.7,"pith_summary":"This paper asks when a group can carry two paratopological group topologies that complement each other in the lattice of topologies: two topologies are T1-complementary when their intersection is the cofinite topology (whose nonempty open sets omit only finitely many points) and their union generates the discrete topology. The main answer is negative for a broad class: if $(G,\\tau)$ is an infinite Hausdorff SD-paratopological group, meaning $\\tau$ admits a coarser saturated topology $\\gamma$ with the same dense subsets, then no Hausdorff paratopological group topology $\\sigma$ can be T1-complementary to $\\tau$. Because the Sorgenfrey line, the real line with the half-open interval topology, is saturated, this settles an open problem in the negative: the Sorgenfrey line has no T1-complementary Hausdorff paratopological group topology. The paper also builds structural tools, submaximal paratopological group topologies and PT-sequences/PT-filters, that characterize when a topology is transversal and when a sequence-determined paratopological group admits a transversal partner. If correct, the results delimit where complementary paratopological group topologies can exist: complementarity is rare, while transversality, for sequence-determined groups, is common.","feed_headline":"Sorgenfrey line has no complementary paratopological partner","feed_subtitle":"The half-open interval topology on R cannot be T1-complemented by another Hausdorff paratopological topology.","key_machinery":"The proofs rest on three mechanisms. (1) The SD-paratopological condition: $(G,\\tau)$ is SD if some coarser saturated paratopological topology $\\gamma$ has exactly the same dense subsets as $\\tau$; saturation means every neighborhood $U$ of the identity has $U^{-1}$ with nonempty interior, and this is the property that converts a hypothetical T1-complement into a dense-set contradiction. (2) The submaximal paratopological group topology $PM_G$, defined as the infimum of all maximal paratopological group topologies on $G$. Theorem 4.2 shows $PM_G$ is a topological group, and Theorem 4.6 uses it as a measuring stick: a non-discrete paratopological group topology $\\tau$ admits a transversal paratopological group topology exactly when $\\tau$ is not below $PM_G$. (3) PT-filters and PT-sequences: a filter or sequence is PT if some paratopological group topology makes it converge to the identity, and $P(G\\mid\\phi)$ is the strongest such topology. Theorem 5.6 gives an explicit neighborhood base at 0, the family $\\Sigma(\\phi^{*})$ of finite-fold sums of tails, and Theorem 5.9 characterizes PT-sequences by asking that each nonzero element be excluded from the finite-sum sets $A(k,m)$. Theorem 5.14 shows $P(G\\mid\\{a_n\\})$ is sequential, and Theorem 5.15 shows that for a nontrivial T-sequence it contains a closed copy of the Arens space $S_2$, hence is not Fréchet–Urysohn.","core_discovery":"The core discovery is that T1-complementarity is impossible for a wide class of paratopological groups, and that transversality can be characterized by a single canonical topology. Theorem 3.3 proves that if $(G,\\tau)$ is an infinite Hausdorff SD-paratopological group, then no Hausdorff paratopological group topology $\\sigma$ is T1-complementary to $\\tau$; the proof shows every $\\sigma$-neighborhood of the identity is dense in the coarser saturated topology $\\gamma$, so it meets every $\\tau$-neighborhood infinitely often, while transversality would require two neighborhoods whose intersection is the singleton identity. Corollary 3.4 applies this to the Sorgenfrey line and answers an open problem negatively. Theorem 3.6 adds a countable-group dichotomy: for a countable infinite group, the absence of T1-independent Hausdorff paratopological group topologies is equivalent to the absence of a transversal T1-independent pair. Theorem 4.6 gives the transversality criterion $(G,\\tau)\\in PTrans$ iff $\\tau$ is not below the submaximal paratopological group topology $PM_G$, and Theorem 4.11 shows transversality lifts from a central subgroup to the ambient non-discrete group. Sections 5 and 6 introduce PT-filters and PT-sequences, characterize when a paratopological group is determined by one, prove that every Hausdorff PT-sequence-determined topology is transversal, show the strongest T-sequence-determined topology admits no T1-complementary Hausdorff paratopological group topology, and prove that a countable group determined by a PT-sequence, when its associated strongest topology is Hausdorff, admits a transversal PT-sequence-determined topology.","pith_inferences":["A natural next test, not addressed in the paper, is whether saturation alone—without the extra dense-set condition in the definition of SD—already blocks T1-complementarity for every infinite Hausdorff paratopological group; the paper proves the obstruction only for SD groups, though the Sorgenfrey line is saturated.","Because Theorem 5.9 reduces PT-sequencehood to a checkable finite-sum condition, one can search for explicit transversal pairs $P(G\\mid\\{a_n\\})$ and $P(G\\mid\\{b_n\\})$ in concrete countable groups such as $\\mathbb{Z}$, using Fibonacci-type or other fast-growing sequences.","If the countable-group equivalence of Theorem 3.6 extended to uncountable groups, then T1-complementarity would reduce entirely to T1-independence at every cardinality; the countable-weight thinning assumption makes that extension nontrivial and testable."],"forward_implications":["The Sorgenfrey line, and more generally every infinite Hausdorff saturated paratopological group, cannot be part of a T1-complementary pair of Hausdorff paratopological group topologies.","A non-discrete paratopological group topology $\\tau$ is transversal exactly when $\\tau$ is not contained in the submaximal paratopological group topology $PM_G$, so transversality becomes a comparison with a single canonical topology.","If a non-discrete paratopological group contains a central subgroup that is either discrete non-minimal or admits a transversal paratopological group topology, then the ambient group admits a transversal paratopological group topology.","Every Hausdorff paratopological group topology on an infinite group determined by a PT-sequence is transversal, while the strongest topology determined by a T-sequence admits no T1-complementary Hausdorff paratopological partner.","For countable infinite groups, the absence of T1-independent Hausdorff paratopological group topologies is equivalent to the absence of a transversal T1-independent pair, and a countable group has a non-discrete T1-paratopological group topology exactly when it has a nontrivial PT-sequence."],"supporting_citations":[{"why":"It poses the open Problem 10 about the Sorgenfrey line and supplies the definitions of transversal, T1-independent and T1-complementary topologies used throughout.","marker":"[5]"},{"why":"It establishes the lattice-theoretic framework for complementary group topologies and proves that no Hausdorff group topology admits a T1-complement, motivating the split into transversality and T1-independence.","marker":"[6]"},{"why":"It supplies the theorem that every Hausdorff paratopological group topology on a countable group contains a coarser one of countable weight, the load-bearing step in Theorem 3.6.","marker":"[13]"},{"why":"It provides the theory of topologies determined by sequences, including T-sequences and PT-sequences, together with the key facts about P(G|{a_n}) and T(G|{a_n}) used in Sections 5 and 6.","marker":"[11]"},{"why":"It gives Proposition 5.19, which derives countable compactness and the absence of nontrivial convergent sequences from T1-independence of a sequential topology, used in Theorem 5.20.","marker":"[15]"},{"why":"It supplies the result, used at the end of Theorem 5.20, that an ω-narrow paratopological group is saturated.","marker":"[1]"},{"why":"It is referenced in Theorem 3.7 to reduce the coarser topologies on a countable Abelian group to Hausdorff topological group topologies.","marker":"[9]"}],"fun_headline_variants":["Sorgenfrey line defies T1-complementarity","No T1-complement for Sorgenfrey topology","Transversality criterion: submaximal topology is key","Central subgroup transversality lifts to whole group","PT-sequences characterize transversal paratopological groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every Hausdorff paratopological group topology on a countable group can be thinned to a coarser Hausdorff paratopological group topology with only countably many basic open sets; if that thinning is impossible for some countable paratopological group, the equivalence in Theorem 3.6 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Sorgenfrey line defies T1-complementarity","No T1-complement for Sorgenfrey topology","Transversality criterion: submaximal topology is key","Central subgroup transversality lifts to whole group","PT-sequences characterize transversal paratopological groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1617,"prompt_tokens":1203,"completion_tokens":414,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":819,"completion_tokens_details":{"reasoning_tokens":338}},"tokens_in":819,"tokens_out":414,"duration_ms":5021,"temperature":1.0,"reasoning_tokens":338,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:53:29.291852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a Hausdorff paratopological group topology $\\sigma$ on $\\mathbb{R}$ such that, with $\\tau$ the Sorgenfrey topology, $\\tau\\cap\\sigma$ is the cofinite topology and $\\tau\\vee\\sigma$ is the discrete topology; such a pair would refute Corollary 3.4 and, with it, Theorem 3.3.","supporting_citations":[{"cited_title":"B/suppress laszczyk, M","cited_arxiv_id":null,"evidence_quote":"It poses the open Problem 10 about the Sorgenfrey line and supplies the definitions of transversal, T1-independent and T1-complementary topologies used throughout."},{"cited_title":"Dikranjan, M","cited_arxiv_id":null,"evidence_quote":"It establishes the lattice-theoretic framework for complementary group topologies and proves that no Hausdorff group topology admits a T1-complement, motivating the split into transversality and T1-independence."},{"cited_title":"Shakhmatov, Condensations of universal topological algebras preservi ng continuity of operations and decreasing weights, Moscow Univ","cited_arxiv_id":null,"evidence_quote":"It supplies the theorem that every Hausdorff paratopological group topology on a countable group contains a coarser one of countable weight, the load-bearing step in Theorem 3.6."},{"cited_title":"Protasov, E","cited_arxiv_id":null,"evidence_quote":"It provides the theory of topologies determined by sequences, including T-sequences and PT-sequences, together with the key facts about P(G|{a_n}) and T(G|{a_n}) used in Sections 5 and 6."},{"cited_title":"Tkachenko, I","cited_arxiv_id":null,"evidence_quote":"It gives Proposition 5.19, which derives countable compactness and the absence of nontrivial convergent sequences from T1-independence of a sequential topology, used in Theorem 5.20."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the result, used at the end of Theorem 5.20, that an ω-narrow paratopological group is saturated."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is referenced in Theorem 3.7 to reduce the coarser topologies on a countable Abelian group to Hausdorff topological group topologies."}],"review_version":1}