{"id":"73e060fc-1ab7-4580-b766-713219a06578","arxiv_id":"2411.15081","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the kernel Q_E*(X) of the double-direction equivalence preserving transformation semigroup, the paper proves rank = max(2, product of equivalence class sizes), an isomorphism criterion, and the count of maximal subsemigroups.","lead":"The paper studies functions that preserve and reverse a partition of a set, and it computes how many functions are needed to generate the whole collection, when two such collections are isomorphic, and how many largest proper pieces each collection has. A generalist might care because these collections are the function-semigroup analogues of symmetric groups, and the answers turn out to be simple formulas in the sizes of the partition classes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4.4 as stated is false for infinite X: the symmetric factor Sym(X/E) is not generated by two elements.","rationale":"The reader's weakest assumption was the import of structural facts from the authors' earlier preprint [9]. That is a legitimate concern, but I found a more direct correctness problem in the stated central claim. Corollary 4.4 has no finiteness hypothesis, and the assertion that every symmetric group on a set of size at least 2 has rank 2 is simply false for infinite sets. The failure is load-bearing because the corollary is included among the paper's strongest claims and is used as the rank theorem. The counterexample with X = N and one two-element E-class plus countably many singleton classes gives X/E countably infinite and m = 2; the paper's Proposition 3.4 then forces Q_E*(X) to project onto Sym(N), so a two-element generating set would give a two-element generating set of Sym(N), which is impossible since Sym(N) is uncountable. Thus rank(Q) > 2. This is not an artifact of the imported results: it follows from the paper's own decomposition. The intended finite version of the result is plausibly salvageable, and the formula max{2,m} happens to be correct for the finite edge cases because m >= 2 when E is non-identity; the main required revisions are to add the finite hypothesis and to correct the proof's treatment of symmetric-group ranks. Example 6 also contains a false minimal generating set for S3, since {alpha1, alpha7} generates only a copy of C2. Because these are concrete, fixable defects rather than a collapse of the finite case, I would keep the reader's conditional verdict rather than move to acceptance or rejection.","tokens_in":12304,"tokens_out":21784,"duration_ms":232265,"concrete_test":"Run the counterexample: let X = N and let E have classes {0,1}, {2}, {3}, ... . Then X/E has cardinality aleph0 and m = 2. Verify that Q_E*(X) is isomorphic to Sym(N) x E_2 as the paper claims, and that any generating set of this product projects to a generating set of Sym(N). Since Sym(N) is uncountable and any two-generated subgroup is countable, no generating set of size 2 exists; conclude that rank(Q) > 2, contradicting max{2,m} = 2. Equivalently, add the hypothesis 'X finite' to Corollary 4.4 and check that this counterexample fails only because X is infinite.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 4.4 is stated for an arbitrary nonempty set X, with no finiteness hypothesis. Its proof invokes the claim that the symmetric group on any set Y with |Y| >= 2 has rank 2. That claim is false for infinite Y. For Y = N, Sym(N) has cardinality 2^{aleph0}, while a group generated by a set of cardinal lambda has cardinality at most max(lambda, aleph0); in particular, no finite or countably infinite generating set can exist. Now take X = N and E the partition {{0,1},{2},{3},...}. Then X/E is countably infinite and m = 2. By Proposition 3.4, Q_E*(X) is isomorphic to Sym(N) x E_2. Projection onto the first factor is a surjective homomorphism, so any generating set of Q would project to a generating set of Sym(N); therefore rank(Q) cannot be 2. This is a concrete counterexample to Corollary 4.4 as written. Even in the finite case, the proof's blanket 'rank 2' assertion mishandles Sym_2 and Sym_1, whose ranks are 1 and 0, although max{2,m} accidentally agrees when m >= 2. A separate concrete error appears in Example 6, where {alpha1, alpha7} is asserted to be a minimal generating set of S3; alpha7 is a transposition, so it generates only the two-element subgroup.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the subsemigroup Q_E*(X) of the full transformation semigroup T(X) consisting of equivalence-preserving transformations whose image meets each E-class in exactly one point. Building on the earlier preprint [9], the authors identify Q_E*(X) with the kernel of the regular part of T_E*(X) and recall that it is a right group, i.e., a disjoint union of pairwise isomorphic symmetric groups. The main new results are: an isomorphism criterion for Q_E*(X) and Q_F*(Y) in terms of the cardinalities of the quotient sets and the products of the class sizes (Theorem 3.5); a rank formula for Q_E*(X) (Corollary 4.4); and a description and count of all maximal subsemigroups of Q_E*(X) when X is finite (Theorem 5.6 and the discussion following it). The proofs are mostly elementary and use the decomposition Q_E*(X) ≅ S_{X/E} × E(Q_E*(X)).","tokens_in":12628,"tokens_out":17231,"duration_ms":172654,"significance":"If the finite versions of the results are correct, the paper gives a clean structural reduction: rank and maximal subsemigroup questions for Q_E*(X) are reduced to questions about a symmetric group and a right zero semigroup. The isomorphism theorem is natural and the counting formula in Section 5 is explicit and checkable. A notable strength is the worked Example 6, which is intended to illustrate the rank computation and the maximal subsemigroup classification. The main reservations are the overstatement of Corollary 4.4 to infinite sets, a concrete generating-set error in Example 6, and the fact that the whole framework rests on structural facts imported from the corresponding author's unpublished preprint [9].","major_comments":[{"comment":"The statement is false for infinite X. The proof invokes the claim that the symmetric group on any set Y with |Y| ≥ 2 has rank 2; this is false for infinite Y. For example, take X = N and E = {{0,1},{2},{3},...}. Then |X/E| = ℵ0 and m = 2, so by Proposition 3.4, Q_E*(X) ≅ Sym(N) × E_2. Projection onto the first factor is a surjective homomorphism, so a 2-element generating set for Q_E*(X) would yield a 2-element generating set for Sym(N). But Sym(N) has cardinality 2^{ℵ0}, while a group generated by a finite set is countable. Corollary 4.4 should be restricted to finite X, and the proof should handle S_1 and S_2 separately; with those changes the finite statement appears correct.","section":"§4, Corollary 4.4"},{"comment":"The set G = {α1, α7} is not a minimal generating set of H_{α1}. The displayed computation α7^2 = α1 already shows α1 is generated by α7, and since α7 is a transposition (α7ψ = (1 2)), ⟨α7⟩ has exactly two elements and cannot generate H_{α1}, which is isomorphic to S3. Consequently the displayed set {α2,...,α7} is not established as a generating set of Q_E*(X), and the claim that it is a minimal generating set is unsupported. The example should use a genuine minimal generating pair for S3, such as a transposition together with a 3-cycle, and then apply Theorem 4.3.","section":"§6, Example 6"},{"comment":"The main results all rely on structural facts imported from the authors' earlier preprint [9]: that Q_E*(X) is the kernel and a right group of Reg(T), that each H-class is isomorphic to S_{Xα}, and the characterization of idempotents. These facts are load-bearing for Proposition 3.4 and therefore for Theorem 3.5, Corollary 4.4, and Section 5. Since [9] is an arXiv preprint rather than a peer-reviewed publication, the revision should either prove these facts or cite a published source for them.","section":"§2 (Corollary 2.2, Theorem 2.3, Lemma 2.9; used in Proposition 3.4)"}],"minor_comments":[{"comment":"The abstract contains grammatical and typographical slips: 'Defined the subsemigroup' should be 'Define the subsemigroup', and the condition 'A ∩ Xα ≠ /0' contains a stray '/0' that should be '∅'.","section":"Abstract"},{"comment":"The expression 'α_2^7 = (4,1,6)^2 = (1,4,6) = α_1' appears to be a typographical error for 'α_7^2'; the computation itself is correct.","section":"§6, Example 6"},{"comment":"The phrase 'symmetric group of order n' should be 'symmetric group of degree n', since the order of S_n is n!, not n.","section":"§5"},{"comment":"The proof of the case |G| ≤ |E(S)| uses the expression |E(S)| − |G|, which is not meaningful for infinite cardinals without comment. Since the intended application is finite, the theorem should either be stated for finite S or supplemented with a short cardinal arithmetic remark.","section":"§4, Theorem 4.3"}],"recommendation":"major_revision","confidential_remarks":"The finite results appear sound and fit the journal's scope. The revision should correct the overbroad infinite statement of Corollary 4.4, replace the erroneous generating set in Example 6, and address the heavy dependence on the corresponding author's unpublished preprint [9]. These issues are fixable within the manuscript's scope, so I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper computes the rank, isomorphism type, and maximal subsemigroup structure of Q_E*(X), the kernel of the regular part of the double-direction equivalence-preserving transformation semigroup. What's actually new: the rank formula (finite case), the isomorphism criterion, and the count of maximal subsemigroups. These are not in the cited prior work. They are reasonably direct consequences of the right-group decomposition imported from the corresponding author's preprint [9], but they are genuine contributions and the paper is clearly organized.\n\nThe finite results look right. The isomorphism theorem, Theorem 3.5, follows cleanly from the right-group decomposition and the fact that the idempotent count is the product of the class sizes. The maximal subsemigroup characterization in Theorem 5.6 is plausible and the count s_n + m is consistent with the known theory. I did not find a load-bearing error in those arguments.\n\nThe soft spots are real but localized. Corollary 4.4 is stated for an arbitrary nonempty set X, and that is wrong. For X = N with E partitioning off {0,1} and all other classes singletons, X/E is countably infinite and m = 2, but any generating set of Q_E*(X) projects to a generating set of Sym(N), which is not 2-generated. So the corollary needs a finiteness hypothesis. The proof also asserts that Sym(Y) has rank 2 for every |Y| >= 2, which is false for |Y| = 2 and for infinite Y; in the finite case the formula still holds because m >= 2, but the argument has to be patched. Example 6 is genuinely wrong: {alpha1, alpha7} is said to be a minimal generating set of H_e, but alpha7 is a transposition and generates only a subgroup of order 2. The later claim that the six-element set {alpha2,...,alpha7} generates Q_E*(X) therefore needs re-examination — as written, it does not generate the H-class. This is an example, not the main theorem, but it should be fixed.\n\nThe reliance on the author's own preprint [9] for the right-group structure, the idempotent characterization, and the H-class descriptions is a caution. It is not circular, but it is load-bearing and should be stated explicitly; ideally the authors would give self-contained proofs or have [9] formally published.\n\nBottom line: this is a competent specialist paper with a correct core in the finite case, and the errors are fixable. It deserves a serious referee, but the referee should require the Corollary 4.4 statement to be corrected, the rank argument repaired, and Example 6 redone.","headline":"Solid finite-rank and isomorphism results for Q_E*(X), but the infinite version of Corollary 4.4 is false and Example 6 needs correction.","tokens_in":13117,"tokens_out":7566,"would_cite":false,"duration_ms":69974,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M17","20M19","20M20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that Q_E^*(X), the kernel of the regular part of the double-equivalence-preserving transformation semigroup, is a right group isomorphic to S_{X/E} × E(Q_E^*(X)), and that this split determines its rank, isomorphism type…","keywords":["transformation semigroup","equivalence relation","right group","rank","maximal subsemigroup","kernel","idempotent","symmetric group"],"falsifier":"For a small finite example such as X = {1,2,3,4,5,6} partitioned into classes of sizes 3, 2, and 1, the paper predicts |Q_E^*(X)| = 36, rank = 6, and exactly 10 maximal subsemigroups (s_3 = 4 for S_3, plus m = 6); a direct exhaustive enumeration of all subsemigroups of this 36-element semigroup that finds a different minimal generating size, a different number of maximal subsemigroups, or a maximal subsemigroup not of the stated form would refute the corresponding theorem.","tokens_in":12092,"feed_emoji":"🧮","tokens_out":8396,"duration_ms":81848,"temperature":0.7,"pith_summary":"This paper studies the semigroup Q_E^*(X) of transformations of a set X that preserve an equivalence relation E in both directions, collapse each E-class to a single point, and whose image meets every E-class. The paper's central claim is that this semigroup is a right group — a disjoint union of groups with a right-zero multiplication between them — and that it splits as a direct product of the symmetric group on the set of E-classes and the right-zero semigroup of its idempotents. From this split, the paper derives the rank (minimum size of a generating set), a complete isomorphism classification, and a count of maximal subsemigroups for finite X. A reader should care because the result reduces a semigroup defined by a double equivalence condition to two simple invariants: the number of equivalence classes and the product of their sizes.","feed_headline":"Rank of a transformation semigroup equals product of class sizes","feed_subtitle":"New split into a symmetric group and a right-zero semigroup fixes generators, isomorphisms, and maximal subsemigroups.","key_machinery":"The load-bearing mechanism is the decomposition Q_E^*(X) ≅ S_{X/E} × E(Q_E^*(X)), obtained by combining the classical right-group theorem with the fact, imported from the earlier preprint [9], that each H-class of Q_E^*(X) is the symmetric group on a cross-section of the partition X/E. The second factor E(Q_E^*(X)) is a right-zero semigroup — multiplication satisfies x y = y — and its elements are exactly the maps sending each E-class into itself; choosing one image point per class gives a bijection with the Cartesian product of the classes, so |E(Q_E^*(X))| = m. This split reduces every rank, isomorphism, and maximal-subsemigroup question to separate problems in a symmetric group and in a right-zero semigroup.","core_discovery":"The central discovery is a structure theorem (Proposition 3.4): Q_E^*(X) is isomorphic to the direct product S_{X/E} × E(Q_E^*(X)). Every element can be identified with a permutation of the equivalence classes together with an independent choice of one representative in each class, and the H-class of an element is the symmetric group on the chosen representatives. Consequently, Corollary 4.4 gives rank(Q_E^*(X)) = max{2, m} where m is the product of the sizes of the E-classes; Theorem 3.5 classifies Q_E^*(X) up to isomorphism by the pair (|X/E|, product of class sizes); and Theorem 5.6 with Corollary 5.7 describes every maximal subsemigroup as either H × E(Q_E^*(X)) with H a maximal subgroup of the symmetric factor, or S_{X/E} × F with F obtained by deleting one idempotent. The paper also records that Q_E^*(X) is a group only when E is the identity relation.","pith_inferences":["Not in the paper, but the same decomposition suggests that for infinite X the rank should be the maximum of the rank of S_{X/E} and the cardinal m, with products and maxima interpreted as cardinals; this extension is not proved here.","Beyond the paper, the classification by the pair (n, m) implies that questions about congruences, subsemigroup lattices, or automorphism groups of these semigroups can be studied entirely in the product S_n × E_m rather than in transformation semigroups.","Since the earlier preprint states that every right group embeds into some Q_E^*(X), these finite kernels can serve as concrete test cases for general conjectures about right-group generation and maximal subsemigroups, with the symmetric factor replaced by any finite group G."],"forward_implications":["Because rank(Q_E^*(X)) = m for every nontrivial E, a minimal generating set has exactly as many elements as the product of the E-class sizes; in the worked example with class sizes 3, 2, and 1, six elements generate the whole 36-element semigroup.","The isomorphism theorem means two such kernels are isomorphic exactly when |X/E| and the product of class sizes match, so no finer information about the individual classes matters.","Every maximal subsemigroup of a finite Q_E^*(X) either keeps the full symmetric factor and deletes one idempotent's entire H-class, or keeps all idempotents and replaces the symmetric factor by one of its maximal subgroups.","The total number of maximal subsemigroups of a finite Q_E^*(X) is s_n + m, where n = |X/E| and s_n is the number of maximal subgroups of the symmetric group S_n, so the enumeration reduces to a known quantity for symmetric groups."],"supporting_citations":[{"why":"Supplies the imported structural facts: Q_E^*(X) is the kernel of Reg(T), is a right group, its H-classes are symmetric groups on cross-sections, and its idempotents are exactly the maps sending each E-class into itself.","marker":"[9]"},{"why":"Provides the classical theorem that a right group is a direct product of a group and a right-zero semigroup, plus the Green's relations background used throughout.","marker":"[1]"},{"why":"Introduced the double-direction equivalence-preserving semigroup T_E^*(X) and its regularity theory, which underlies the definition of Q_E^*(X) and part of Theorem 2.6.","marker":"[2]"},{"why":"Supplies Lemma 4.3, used to characterize maximal subsemigroups of a right-zero semigroup and hence to prove Theorem 5.6.","marker":"[12]"},{"why":"Provides the sequence s_n of numbers of maximal subgroups of the symmetric group S_n, used for the final count of maximal subsemigroups.","marker":"[7]"},{"why":"Supplies the fact that a right group is a disjoint union of isomorphic groups and is needed for the isomorphism between H-classes and the product structure used in Section 5.","marker":"[5]"},{"why":"Supplies the standard result that symmetric groups are isomorphic exactly when their underlying sets have the same cardinality, used in the isomorphism theorem.","marker":"[3]"},{"why":"Supplies the theorem that a direct product of semigroups is a group exactly when both factors are groups, used in Proposition 2.8.","marker":"[4]"}],"fun_headline_variants":["Rank of kernel: max of 2 and product of class sizes","Kernel semigroup splits into symmetric group and right-zero","Isomorphism theorem and maximal subsemigroups of kernel","Kernel is symmetric group times right-zero semigroup"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument leans on an earlier result, cited as [9], that Q_E^*(X) is the kernel of the regular part, is a right group, has H-classes isomorphic to symmetric groups on cross-sections, and has idempotents exactly the maps sending each E-class into itself; if any of these imported facts fails, the decomposition and all three main theorems would need revision.","fun_headline_variants_meta":{"raw":{"variants":["Rank of kernel: max of 2 and product of class sizes","Kernel semigroup splits into symmetric group and right-zero","Isomorphism theorem and maximal subsemigroups of kernel","Kernel is symmetric group times right-zero semigroup"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001175,"raw_usage":{"total_tokens":4913,"prompt_tokens":1059,"completion_tokens":3854,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":675,"completion_tokens_details":{"reasoning_tokens":3786}},"tokens_in":675,"tokens_out":3854,"duration_ms":23907,"temperature":1.0,"reasoning_tokens":3786,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:33:05.066203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small finite example such as X = {1,2,3,4,5,6} partitioned into classes of sizes 3, 2, and 1, the paper predicts |Q_E^*(X)| = 36, rank = 6, and exactly 10 maximal subsemigroups (s_3 = 4 for S_3, plus m = 6); a direct exhaustive enumeration of all subsemigroups of this 36-element semigroup that finds a different minimal generating size, a different number of maximal subsemigroups, or a maximal subsemigroup not of the stated form would refute the corresponding theorem.","supporting_citations":[{"cited_title":"The regular part of transformation semigroups that preserve double direction equivalence relation","cited_arxiv_id":"2306.08932","evidence_quote":"Supplies the imported structural facts: Q_E^*(X) is the kernel of Reg(T), is a right group, its H-classes are symmetric groups on cross-sections, and its idempotents are exactly the maps sending each E-class into itself."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical theorem that a right group is a direct product of a group and a right-zero semigroup, plus the Green's relations background used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the double-direction equivalence-preserving semigroup T_E^*(X) and its regularity theory, which underlies the definition of Q_E^*(X) and part of Theorem 2.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 4.3, used to characterize maximal subsemigroups of a right-zero semigroup and hence to prove Theorem 5.6."},{"cited_title":"Mitchell and W","cited_arxiv_id":null,"evidence_quote":"Provides the sequence s_n of numbers of maximal subgroups of the symmetric group S_n, used for the final count of maximal subsemigroups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fact that a right group is a disjoint union of isomorphic groups and is needed for the isomorphism between H-classes and the product structure used in Section 5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard result that symmetric groups are isomorphic exactly when their underlying sets have the same cardinality, used in the isomorphism theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that a direct product of semigroups is a group exactly when both factors are groups, used in Proposition 2.8."}],"review_version":1}