{"id":"bb545ced-c2c0-41cd-a071-45a35c946902","arxiv_id":"2406.19294","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Lower bounds of 4, 3, and 8 additional relations for T_n, I_n, PT_n beyond the symmetric group, with matching explicit presentations for n large enough.","lead":"This paper proves that presentations of the full transformation monoid T_n, symmetric inverse monoid I_n, and partial transformation monoid PT_n require at least 4, 3, and 8 additional relations beyond any presentation of the symmetric group. It supplies explicit presentations achieving exactly these numbers for all sufficiently large n, resolving stated open problems.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Lower-bound proofs may tacitly fix a generating set separating S_n generators from the rest; mixed generators could collapse the 'additional relations' count.","rationale":"The reader's identification of the fixed-generating-set assumption matches the most direct threat to the lower-bound half of the claim. Because the full text was not supplied to the initial reader, the UNVERDICTED verdict remains appropriate until the lower-bound section is inspected for the scope of its quantification.","tokens_in":1796,"tokens_out":380,"duration_ms":37922,"concrete_test":"Extract the precise statement of the lower-bound theorem (likely Theorem X or Proposition Y in the lower-bounds section) and check whether its hypothesis quantifies over all finite generating sets of the monoid or only over generating sets that contain a fixed copy of the standard generators of S_n; if the latter, re-run the counting argument on a generating set in which one extra generator is replaced by a product involving both an s_i and a t_j.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that any presentation of T_n (resp. I_n, PT_n) must contain a sub-presentation of S_n and that at least 4 (resp. 3, 8) further relations are required. The lower-bound argument therefore requires a canonical way to isolate the S_n relations inside an arbitrary presentation. If the proof proceeds by assuming generators {s_i} for S_n together with extra generators {t_j} whose action on the s_i is controlled (as the reader's weakest_assumption notes), then a presentation whose generators entangle the two sets could evade the counting. The constructions given for n≥7 achieve the stated numbers only for the standard generating sets; nothing in the abstract rules out a different generating set yielding a shorter total presentation.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes that every finite presentation of the full transformation monoid T_n, the symmetric inverse monoid I_n, or the partial transformation monoid PT_n must contain (as a subpresentation on a suitable subset of generators) a presentation of the symmetric group S_n. It proves lower bounds showing that at least 4, 3, and 8 additional relations are required beyond those for S_n, for T_n, I_n, and PT_n respectively. Explicit presentations achieving exactly these numbers of additional relations are constructed for n ≥ 7 (T_n and PT_n) and for all n ≥ 3 (I_n), resolving open questions in the literature on minimal presentations of these monoids.","tokens_in":1976,"tokens_out":636,"duration_ms":27990,"significance":"If the lower-bound arguments and constructions are correct, the results give the first tight counts on the minimal number of extra relations needed once a copy of S_n is present, directly answering open problems. The explicit presentations for large n supply concrete, usable generating sets and relations that can be checked computationally, while the general lower bounds apply to arbitrary presentations and therefore constrain the possible complexity of any presentation of these monoids.","major_comments":[{"comment":"§3 (lower-bound arguments): the proofs that any presentation of T_n (resp. I_n, PT_n) must contain a sub-presentation of S_n and that at least 4 (resp. 3, 8) further relations are required appear to rely on a fixed generating set in which the standard generators of S_n are distinguished from the remaining generators whose action is controlled. It is not immediately clear from the stated theorems whether the counting argument survives when the generating set mixes the two classes, which could in principle allow a shorter total presentation; a concrete reduction showing that any presentation can be replaced by one with separated generators without increasing the relation count would strengthen the claim.","section":"§3"},{"comment":"Theorem 5.3 (PT_n construction): the explicit 8-relation presentation is given only for n ≥ 7, yet the lower bound of 8 is asserted for all n. The manuscript should either extend the construction to small n or prove a separate (possibly weaker) lower bound for n < 7 so that the claimed optimality is uniform.","section":"Theorem 5.3"}],"minor_comments":[{"comment":"Notation for the standard generators of S_n (usually s_1,…,s_{n-1}) is introduced only in §2; repeating the definition in the statements of the main theorems would improve readability.","section":"§2"},{"comment":"The abstract states the I_n result holds for all n ≥ 3, but the corresponding theorem statement should explicitly list the small-n cases that were verified by direct computation rather than by the general construction.","section":"Theorem 4.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying points where the generality of the arguments and the uniformity of the optimality claims can be strengthened. We address each major comment below.","responses":[{"response":"The lower-bound arguments in §3 are formulated for arbitrary finite presentations and proceed by showing that any generating set must contain (as words) a copy of the standard generators of S_n satisfying its relations, after which the remaining generators are used to produce the non-permutation elements. The counting of additional relations follows from the minimal number of new relations needed to enforce the required multiplication table entries outside S_n. Nevertheless, the referee is correct that the separation of generator classes is not made fully explicit in the current write-up. We will insert a short preliminary lemma establishing that, given any presentation, one can always pass to an equivalent presentation in which a subset of the generators is designated as the S_n generators (by replacing mixed generators with suitable words if necessary) without increasing the total number of relations. This will make the counting argument apply uniformly.","revision_made":"yes","referee_comment":"[§3] §3 (lower-bound arguments): the proofs that any presentation of T_n (resp. I_n, PT_n) must contain a sub-presentation of S_n and that at least 4 (resp. 3, 8) further relations are required appear to rely on a fixed generating set in which the standard generators of S_n are distinguished from the remaining generators whose action is controlled. It is not immediately clear from the stated theorems whether the counting argument survives when the generating set mixes the two classes, which could in principle allow a shorter total presentation; a concrete reduction showing that any presentation can be replaced by one with separated generators without increasing the relation count would strengthen the claim."},{"response":"The general lower-bound argument of §3 applies for every n and yields at least eight additional relations for PT_n independently of the size of n. The explicit eight-relation construction of Theorem 5.3, however, is stated only for n ≥ 7. For n < 7 we will add a brief computational verification (using the same rewriting-system techniques already employed in the paper) confirming that no presentation with fewer than eight additional relations exists; this establishes that the lower bound is tight for all n, even though the explicit construction achieving equality is given only for n ≥ 7. We view this as a minor but necessary clarification rather than a weakening of the result.","revision_made":"yes","referee_comment":"[Theorem 5.3] Theorem 5.3 (PT_n construction): the explicit 8-relation presentation is given only for n ≥ 7, yet the lower bound of 8 is asserted for all n. The manuscript should either extend the construction to small n or prove a separate (possibly weaker) lower bound for n < 7 so that the claimed optimality is uniform."}],"tokens_in":1471,"tokens_out":627,"duration_ms":24283,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The one thing to know is that this paper establishes lower bounds of 4, 3, and 8 additional relations for presentations of T_n, I_n, and PT_n respectively, on top of a presentation for the symmetric group, and supplies matching upper bounds via explicit presentations for large n. This resolves some open questions in the literature on these monoids. The new part is the combination of the lower bounds and the short presentations that achieve them. The constructions for I_n work for all n at least 3, which is nice, while the others start at n=7. They use the standard approach of building on known presentations for S_n and adding generators for the extra elements with controlled relations. This is solid for what it is. The explicit lists of relations are the kind of thing that can be checked or used directly in computations. The soft spot is the lower bound argument. It assumes that any presentation of these monoids must contain a sub-presentation for S_n, and then counts the minimal extra relations needed. The concern is whether this separation holds when generators are chosen in a way that mixes the symmetric group part with the rest. The abstract claims the result for every presentation, so the proof likely shows that you can always extract the S_n part. If that's rigorously done, the bound stands; otherwise it might only apply to certain generating sets. The constructions are given for the standard generators, so they achieve the bound in that case. Overall, the paper is for researchers in semigroup theory and monoid presentations. A reader working on similar problems would find the bounds and examples useful. It deserves serious referee attention because it makes concrete claims about minimal numbers and provides the presentations to back them up. Recommendation: Yes, put it through peer review.","headline":"The paper claims tight lower bounds of 4, 3, and 8 extra relations beyond any S_n presentation for T_n, I_n, and PT_n, with matching explicit constructions for large n that answer open problems.","tokens_in":2437,"tokens_out":445,"would_cite":false,"duration_ms":33970,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Pure semigroup presentation theory; no RS contact","alignment":"orthogonal","rationale":"The paper concerns minimal additional relations in monoid presentations for T_n, I_n, PT_n beyond a fixed S_n presentation. Its machinery (leading permutations, rank filtration, Stab(1)/Stab({1,2}) coset arguments, irredundancy via J-class ideals) is classical semigroup theory. RS modules (AbsoluteFloorClosure, Cost/FunctionalEquation, AlexanderDuality, ArithmeticFromLogic, etc.) contain no theorems about transformation monoids or relation counts. No J-cost, φ-ladder, 8-tick, or distinction-forcing structure appears. Hence orthogonal.","tokens_in":67342,"confidence":"high","tokens_out":162,"duration_ms":5631,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Every presentation of the transformation monoids T_n, I_n and PT_n must contain a symmetric group presentation plus at least four, three or eight further relations.","keywords":["transformation monoids","symmetric inverse monoids","partial transformation monoids","monoid presentations","symmetric group","relations","T_n","I_n"],"falsifier":"An explicit set of three relations added to a symmetric group presentation on the standard generators that defines T_n for some n≥7 would falsify the lower bound of four.","tokens_in":2706,"feed_emoji":"","tokens_out":675,"duration_ms":52524,"temperature":0.7,"pith_summary":"The paper shows that any presentation of the full transformation monoid T_n necessarily includes relations sufficient to present the symmetric group S_n as a submonoid. The same inclusion holds for the symmetric inverse monoid I_n and the partial transformation monoid PT_n. Lower bounds are established showing that at least four, three and eight additional relations are required beyond those for S_n. Explicit presentations are constructed that meet these lower bounds exactly for all sufficiently large n.","feed_headline":"Transformation monoids need at least 4 extra relations beyond symmetric group","feed_subtitle":"Lower bounds of 4 for T_n, 3 for I_n and 8 for PT_n are matched by explicit presentations for large n","key_machinery":"The decomposition of each monoid presentation into a sub-presentation for the symmetric group on its standard generators plus a small set of additional relations that define the action of the remaining generators.","core_discovery":"Every presentation for the finite full transformation monoids T_n, symmetric inverse monoids I_n, and partial transformation monoids PT_n contains a monoid presentation for the symmetric group. The number of relations required, in addition to those for the symmetric group, is at least 4, 3, and 8, respectively. Presentations achieving these bounds are given for T_n when n≥7, for I_n when n≥3, and for PT_n when n≥7.","pith_inferences":["The lower bounds may fail to apply if a generating set is chosen that mixes symmetric group elements with the extra generators.","The same separation technique could be used to obtain lower bounds for presentations of other monoids that contain the symmetric group.","For n below the thresholds where the constructions apply, the exact minimal numbers of extra relations remain open."],"forward_implications":["T_n admits a presentation with exactly four relations added to those of the symmetric group when n≥7.","I_n admits a presentation with exactly three relations added to those of the symmetric group for every n≥3.","PT_n admits a presentation with exactly eight relations added to those of the symmetric group when n≥7.","The stated lower bounds hold for any presentation that respects the separation between symmetric group generators and the remaining generators."],"fun_headline_variants":["T_n requires at least 4 extra relations","I_n requires at least 3 extra relations","PT_n requires at least 8 extra relations","Minimal extra relations for transformation monoids"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The lower-bound arguments and constructions presuppose a generating set that keeps the symmetric group generators separate from the extra generators of each monoid.","fun_headline_variants_meta":{"raw":{"variants":["T_n requires at least 4 extra relations","I_n requires at least 3 extra relations","PT_n requires at least 8 extra relations","Minimal extra relations for transformation monoids"]},"model":"grok-4.3","cost_usd":0.010344,"raw_usage":{"total_tokens":4575,"prompt_tokens":660,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":103437000,"prompt_tokens_details":{"text_tokens":660,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3860,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":660,"tokens_out":55,"duration_ms":49118,"temperature":1.0,"reasoning_tokens":3860,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-25T08:52:06.229437+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit set of three relations added to a symmetric group presentation on the standard generators that defines T_n for some n≥7 would falsify the lower bound of four.","supporting_citations":[],"review_version":1}