{"id":"1686d243-e8fb-43d5-abe9-a9f54ca821ee","arxiv_id":"2411.14897","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new class of semigroups based on networks is introduced, with an isomorphism theorem relating the semigroups to their networks.","lead":"This paper introduces a new family of semigroups built from \"networks\", which generalize directed graphs by letting an edge connect a set of vertices to another set. The authors claim that each such semigroup determines its underlying network up to isomorphism.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2's vertex-recovery step is invalid: Lemma 5.1(i) makes [A] for every A in T0 maximal, not just [v], so theta need not map vertex idempotents to vertex idempotents.","rationale":"The reader is right to reject the paper, but I would not base the rejection on the confluence criticism exactly as stated. In Proposition 3.2, the alleged missing overlaps where both t1t2 and t2t3 trigger NR3 or both trigger NR5 cannot occur because X is a disjoint union T union T0 union T^-1: for t1t2 to satisfy NR3, the middle generator t2 must lie in T, while for t2t3 to satisfy NR3, t2 would have to lie in T^-1, and similarly for NR5. So the specific counterexample mentioned in the reader's weakest_assumption does not land. The strongest load-bearing problem is internal to Theorem 5.2. Lemma 5.1(i) itself states that every A in T0 gives a maximal idempotent, and T0 contains non-singleton source and range sets. The theorem's proof immediately misreads this as 'vertices only' and uses that misreading to construct the vertex bijection. Since non-vertex maximal idempotents can in principle be permuted with vertex idempotents, the claimed semigroup-theoretic recovery of V is not established. Rejecting the paper is appropriate: the main theorem is the abstract's headline result, and its proof has a concrete, load-bearing gap. The verdict should remain REJECT, with the defect located in Theorem 5.2 rather than in the specific confluence overlap identified by the reader.","tokens_in":22100,"tokens_out":10316,"duration_ms":105277,"concrete_test":"Compute the idempotent poset for the minimal network Gamma=(V,T) with V={a,b,c}, T={t}, s(t)={a,b}, r(t)={c}, using the right normal forms of Theorem 3.5, and list the maximal idempotents under the natural partial order. Lemma 5.1(i) gives [a], [b], [c], and [{a,b}]. Then apply the first paragraph of the proof of Theorem 5.2 to theta = id_QGamma: the asserted bijection from {[v]: v in V} to itself is not defined because theta fixes [{a,b}], a maximal idempotent outside that set. This directly falsifies the 'no other element' assertion in the proof. A stronger check would be to search, by normal-form enumeration for small networks, for an automorphism of QGamma moving [{a,b}] to [a]; such an automorphism would show that vertexhood is not invariant under QGamma-automorphisms and that the proof needs a genuinely different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires an isomorphism QGamma -> QDelta to induce a bijection between V_Gamma and V_Delta. The proof's first reduction asserts that among idempotents, exactly the [v] (v in V) are maximal in E(QGamma), and then restricts theta to {[v]}. But Lemma 5.1(i) actually says maximal idempotents are exactly [alpha alpha^-1] with alpha in T0, i.e. [A] for all A in T0, and T0 contains every source and range set s(t), r(t), not only singletons. For example, take Gamma with V={a,b,c}, T={t}, s(t)={a,b}, r(t)={c}. The maximal idempotents are [a], [b], [c], and [{a,b}]. Thus a semigroup isomorphism may send a non-vertex maximal idempotent [A]_Gamma to a vertex idempotent [v]_Delta or vice versa, so the claimed restriction of theta to {[v]: v in V_Gamma} is not a well-defined bijection. Without an extra semigroup-definable invariant separating V from T0\\V, the construction of phi_V collapses. All subsequent steps — [t]theta=[q], [s(t)]theta={[v]theta: v in s(t)}, and [r(t)]theta similarly — depend on that bijection, so Theorem 5.2 is unsupported. Lemma 5.1(ii) does not repair the gap: it only separates the relation-idempotents [tt^-1] inside E(QGamma)\\{[A]: A in T0}, and that quotient already assumes the problematic [A] have been identified and discarded.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the class of network right ∗-abundant semigroups QΓ associated with a network Γ=(V,T,s,r), where relations may have non-singleton sources and ranges. It presents a semigroup presentation, develops a unique normal form theory, identifies the right ample subsemigroup SΓ and a fundamental inverse subsemigroup RΓ, studies ideals and congruence-free conditions, and states as the main theorem (Theorem 5.2) that Γ≅Δ if and only if QΓ≅QΔ. The claimed converse direction—that the semigroup QΓ is a complete invariant for the network Γ—is the central advertised result of the paper.","tokens_in":22472,"tokens_out":12058,"duration_ms":124669,"significance":"The construction is a natural generalization of graph inverse semigroups, and the paper contains several useful structural results, including a presentation, unique normal forms (Corollary 3.6), a characterization of regular elements (Lemma 3.10), and the right ample subsemigroup SΓ. These parts are of independent interest. However, the main isomorphism theorem is not established: the proof of Theorem 5.2 relies on an incorrect description of the maximal idempotents of QΓ. Since the abstract and introduction advertise exactly this theorem, the paper in its present form cannot be accepted.","major_comments":[{"comment":"The proof of Theorem 5.2 asserts that \"By part (i) of Lemma 5.1, every vertex corresponding [v] in E(QΓ), but no other element of QΓ, is maximal in E(QΓ), with respect to ≤QΓ.\" This is inconsistent with Lemma 5.1(i), which states that an idempotent [αα−1] is maximal in E(QΓ) if and only if α ∈ T0Γ, where T0Γ = VΓ ∪ {s(t), r(t) : t ∈ TΓ}. Thus, whenever a relation has a non-singleton source or range, the maximal idempotents include [s(t)] and [r(t)] in addition to the vertex idempotents [v]. Consequently the restriction of θ to {[v] : v ∈ VΓ} is not known to map into {[v′] : v′ ∈ VΔ}; a semigroup isomorphism may send a non-vertex maximal idempotent of QΓ to a vertex idempotent of QΔ or vice versa. The subsequent construction of the bijection θVΓ : VΓ → VΔ and the equalities [s(t)]θ = {[v]θ : v ∈ s(t)} and [r(t)]θ = {[v]θ : v ∈ r(t)} all depend on this unjustified restriction. The main theorem is therefore unsupported.","section":"5, Theorem 5.2"},{"comment":"Even if the vertex-recovery issue were repaired, the proof that the source and range maps of the induced network isomorphism are preserved is not rigorously written. The text writes [r(t)]θ = {[v]θ : v ∈ r(t)} = [r([t]θ)], where the left-hand side is an element of QΔ while the right-hand side is a set; this conflates elements of QΔ with subsets of VΔ. A correct proof would need to show, once a bijection between vertex idempotents is established, that the idempotent [r(t)]θ equals [r([t]θ)] in QΔ and that this element encodes exactly the set r(t) under that bijection. As written, this step does not provide a formal verification that θ preserves the network source and range maps.","section":"5, Theorem 5.2 (proof, final paragraph)"}],"minor_comments":[{"comment":"The text says \"since t2 = r(t3) ∈ T0\" but the preceding line gives t2 = s(t3); this appears to be a typo and makes the case verification harder to follow.","section":"3, Proposition 3.2, Case 1(a1)"},{"comment":"The assertion that \"it never happens that both t1t2 and t2t3 satisfy the relations among (NR3), (NR4) and (NR5)\" is correct, because the middle symbol would have to belong simultaneously to T and to T−1 or T0, but the justification is not stated; adding one sentence explaining the disjointness of the generator sets would improve readability.","section":"3, Proposition 3.2"},{"comment":"The product formula in Lemma 3.8 is central but its proof is terse, especially the prefix-comparability case; the authors should reorganize the argument by separating the cases β ∈ T0, β ∈ RLP(Γ)\\T0 with μ a prefix of β, and β a prefix of μ, to make the normal-form computations easier to verify.","section":"3, Lemma 3.8"},{"comment":"The notation XAA and XAt1 in the example is not explained: it is not clear whether XAA means words ending with the symbol A or words of the form (word in X)A, and the definition of XA does not immediately clarify this. A short explanation would make the example accessible.","section":"6, Example"},{"comment":"There are several typographical slips, such as \"P R(Γ)\" for \"RP(Γ)\" in the definition of SΓ, and inconsistent spacing in expressions like \"αβ −1\"; a careful copyedit is needed.","section":"Overall"}],"recommendation":"reject","confidential_remarks":"The paper has a genuinely interesting construction, but the central theorem's proof is invalid. The error is not a minor gap: the maximal idempotents of QΓ are exactly [A] for A ∈ T0, not just [v] for v ∈ V. If the isomorphism theorem is to be salvaged, the authors would need a new argument to recover V from E(QΓ), or additional hypotheses on the networks. I recommend rejection of the current version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the construction is genuinely new: QΓ built from networks with set-valued sources and ranges, recovering graph inverse semigroups in the singleton case, with a careful development of right *-abundant structure and a right ample subsemigroup SΓ and fundamental inverse subsemigroup RΓ. Second, the headline result, Theorem 5.2, is not proved as written. The proof claims that among idempotents exactly the [v] (v∈VΓ) are maximal in E(QΓ). Lemma 5.1(i) says the maximal idempotents are exactly [A] for all A∈T0Γ, which includes non-singleton source and range sets, not just singletons. So an isomorphism QΓ→QΔ need not restrict to a bijection from {[v]: v∈VΓ} to {[v']: v'∈VΔ}. The subsequent construction of the vertex map collapses: [s(t)]θ={[v]θ: v∈s(t)} and [r(t)]θ={[v]θ: v∈r(t)} both depend on that bijection. The stress-test example (one relation with s(t)={a,b}, r(t)={c}) makes the failure explicit. This is not a minor gap; it is the main theorem.\n\nThe confluence proof of Proposition 3.2 also has an omission. The case analysis does not treat configurations where both t1t2 and t2t3 trigger (NR3) (e.g., t1=a−1, t2=b, t3=c) or both trigger (NR5). Confluence is load-bearing: it gives the unique normal forms in Theorem 3.5, which support Lemma 3.8, Lemma 3.9, and the L∗-class structure. If it fails, the structure theory built on it is unsupported. I did not find an actual counterexample to confluence, so this may be repairable, but it needs a completed proof.\n\nCredit where due. The definitions are clean and the paper is honest about its scope. The right ∗-abundant and right ample results (Theorem 3.14, Proposition 3.15) follow naturally once the normal form theory is in place. The ideal and congruence-free sections are plausible and contain real content. The paper is not circular and does not overclaim beyond the abstract—except that the abstract's main theorem is unsupported.\n\nWho is this for? Semigroup theorists working on abundant semigroups and on generalizations of graph inverse semigroups. It deserves a serious referee, but the referee should demand a corrected proof of Theorem 5.2 and a completed confluence argument before acceptance. If those repairs succeed, this would be a solid contribution to the literature.","headline":"Genuinely new construction, but the advertised isomorphism theorem is unsupported: Lemma 5.1(i) contradicts the proof's maximal-idempotent claim, and the confluence proof has a gap.","tokens_in":22962,"tokens_out":2441,"would_cite":false,"duration_ms":21007,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20M10","20M05","20M18","05C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that two network right *-abundant semigroups, built from paths in networks that generalize directed graphs, are isomorphic if and only if the networks underneath are isomorphic.","keywords":["network right *-abundant semigroup","networks","graph inverse semigroups","right abundant semigroups","right ample semigroups","confluent rewriting system","semigroup isomorphism","natural partial order"],"falsifier":"To settle the claim, test whether the reduction system is locally confluent in the unlisted three-letter configurations where both adjacent pairs are zero-producing; a single word with two distinct irreducible descendants would break Proposition 3.2. More directly, search small networks for non-isomorphic Γ and Δ whose semigroups QΓ and QΔ are isomorphic, which Theorem 5.2 predicts cannot happen.","tokens_in":21894,"feed_emoji":"🕸️","tokens_out":10123,"duration_ms":94890,"temperature":0.7,"pith_summary":"The paper introduces network right *-abundant semigroups, a class of semigroups with zero constructed from paths in a network, where a network is a generalization of a directed graph in which each relation connects two disjoint non-empty sets of vertices rather than two single vertices. The construction generalizes graph inverse semigroups, which appear when every relation connects singletons. The main claim is a complete classification up to isomorphism: two network right *-abundant semigroups are isomorphic if and only if the networks they are built from are isomorphic. If true, this means the semigroup remembers its network exactly, so questions about isomorphism of such semigroups reduce to questions about isomorphism of networks, and vice versa.","feed_headline":"Two network semigroups are isomorphic if and only if networks are","feed_subtitle":"The paper builds semigroups from generalized directed graphs and shows the algebra records the whole network exactly.","key_machinery":"The load-bearing mechanism is the presentation QΓ = ⟨X : R⟩ with X = T ∪ T0 ∪ $T^{{-1}}$ ∪ {0} and rewriting rules (NR1)–(NR6), together with the confluence of the reduction system (X^+, →) proved in Proposition 3.2. Confluence gives every element a unique normal form $αβ^{{-1}}$ with α a reduced path, β a reduced linear path, and r(α) = r(β). Right *-abundant means every L*-class contains a unique idempotent. Idempotents are exactly [$αα^{{-1}}$] for reduced linear paths α plus zero; comparing them under the natural partial order and locating the maximal ones recovers the vertex set and the relation set of Γ, which is what makes the isomorphism theorem work.","core_discovery":"The central discovery is that the network Γ is a complete invariant for the semigroup QΓ: Theorem 5.2 states Γ ≅ Δ if and only if QΓ ≅ QΔ. The proof reconstructs the network from the semigroup by looking at the natural partial order on idempotents: vertices are exactly the maximal idempotents [v] among all idempotents, and relations are exactly the maximal idempotents [$tt^{{-1}}$] in the semilattice obtained after removing the idempotents coming from T0. Along the way the paper establishes that QΓ is a right *-abundant semigroup with zero, isolates a right ample subsemigroup SΓ and a fundamental inverse subsemigroup RΓ, and shows QΓ is not left abundant in general.","pith_inferences":["The reconstruction of Γ from maximal idempotents suggests that automorphism groups of networks embed into automorphism groups of QΓ, though the paper does not state this consequence.","The same strategy could be tried on other algebraic objects built from higher-order networks, such as path algebras of hypergraph-like structures, to test whether 'the algebra remembers the hypergraph' holds there.","The paper proves the invariant only for QΓ itself; whether the proper quotient QΓ/I or the inverse subsemigroup RΓ also determines Γ remains open."],"forward_implications":["Graph inverse semigroups are exactly the special case where every relation joins two singleton vertices, so the isomorphism theorem covers and extends the graph case.","The isomorphism problem for network right *-abundant semigroups is the same problem as isomorphism of networks: neither problem is harder than the other.","The idempotents of QΓ, ordered naturally, carry enough information to read off the vertex set and the relation set of Γ.","The class properly contains graph inverse semigroups and includes examples where QΓ is not left abundant, so the right *-abundant setting is genuinely wider."],"supporting_citations":[{"why":"Supplies the noetherian/local-confluence criterion used to prove the reduction system is confluent and thus that every element has a unique normal form.","marker":"[7]"},{"why":"Defines the natural partial order on semigroups; the proof of Theorem 5.2 uses maximal idempotents under this order to recover vertices and relations.","marker":"[12]"},{"why":"Introduces graph inverse semigroups, the class that network right *-abundant semigroups properly extend.","marker":"[3]"},{"why":"Provides the presentation and αβ^{-1} normal form for graph inverse semigroups that the network construction generalizes.","marker":"[15]"},{"why":"Supplies the background semigroup theory (regular elements, Green's relations, idempotents) used throughout the structure proofs.","marker":"[11]"},{"why":"Defines abundant and right abundant semigroups and the L* and R* relations that make QΓ a right *-abundant semigroup.","marker":"[9]"}],"fun_headline_variants":["Networks fully determine their right *-abundant semigroups","Networks are complete invariants for their *-abundant semigroups","Semigroup isomorphism for these algebras is network isomorphism","Networks are the sole invariant for these semigroups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the rewriting rules for words being confluent, meaning every word has a single canonical reduced form no matter the order of reductions; if that fails, the unique normal forms used throughout the isomorphism proof collapse.","fun_headline_variants_meta":{"raw":{"variants":["Networks fully determine their right *-abundant semigroups","Networks are complete invariants for their *-abundant semigroups","Semigroup isomorphism for these algebras is network isomorphism","Networks are the sole invariant for these semigroups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000729,"raw_usage":{"total_tokens":3170,"prompt_tokens":755,"completion_tokens":2415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":371,"completion_tokens_details":{"reasoning_tokens":2344}},"tokens_in":371,"tokens_out":2415,"duration_ms":17954,"temperature":1.0,"reasoning_tokens":2344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:45:04.653491+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To settle the claim, test whether the reduction system is locally confluent in the unlisted three-letter configurations where both adjacent pairs are zero-producing; a single word with two distinct irreducible descendants would break Proposition 3.2. More directly, search small networks for non-isomorphic Γ and Δ whose semigroups QΓ and QΔ are isomorphic, which Theorem 5.2 predicts cannot happen.","supporting_citations":[{"cited_title":"V., Otto, F.: String-Rewriting Systems, Springer, 1993","cited_arxiv_id":null,"evidence_quote":"Supplies the noetherian/local-confluence criterion used to prove the reduction system is confluent and thus that every element has a unique normal form."},{"cited_title":"Mitsch.: A natural partial order for semigroups","cited_arxiv_id":null,"evidence_quote":"Defines the natural partial order on semigroups; the proof of Theorem 5.2 uses maximal idempotents under this order to recover vertices and relations."},{"cited_title":"J., Hall, T","cited_arxiv_id":null,"evidence_quote":"Introduces graph inverse semigroups, the class that network right *-abundant semigroups properly extend."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the presentation and αβ^{-1} normal form for graph inverse semigroups that the network construction generalizes."},{"cited_title":"M.: An Introduction to Semigroup Theory","cited_arxiv_id":null,"evidence_quote":"Supplies the background semigroup theory (regular elements, Green's relations, idempotents) used throughout the structure proofs."},{"cited_title":"B.: Abundant semigroups","cited_arxiv_id":null,"evidence_quote":"Defines abundant and right abundant semigroups and the L* and R* relations that make QΓ a right *-abundant semigroup."}],"review_version":1}