Network right * abundant semigroups

Pei Gao; Xueming Ren; Yanhui Wang

REVIEW 2 major objections 5 minor 17 references

Network right * abundant semigroups

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.14897 v1 pith:R34BQC6Q submitted 2024-11-22 math.GR

Yanhui Wang , Pei Gao , Xueming Ren This is my paper · ORCID

classification math.GR MSC 20M1020M0520M1805C20

keywords networkright*-abundantsemigroupnetworksgraphinversesemigroupsabundantampleconfluentrewritingsystemisomorphismnaturalpartialorder

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Credit 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.

Who 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[5, Theorem 5.2] 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.
[5, Theorem 5.2 (proof, final paragraph)] 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.

minor comments (5)

[3, Proposition 3.2, Case 1(a1)] 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.
[3, Proposition 3.2] 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.
[3, Lemma 3.8] 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.
[6, Example] 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.
[Overall] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: QΓ is built from Γ by an explicit presentation and Theorem 5.2 is proved by reconstructing Γ from the idempotent poset, not by assuming the conclusion.

full rationale

The construction of QΓ is an explicit presentation (Definition 3.1) with generators T ∪ T0 ∪ T^{-1} ∪ {0} and relations NR1–NR6. The unique-normal-form result (Theorem 3.5) is proved via the confluence argument in Proposition 3.2, with the one-step triple reductions checked by cases, and via Lemma 2.6, which cites an external rewriting-systems text. No parameter is fitted and no 'prediction' is imported from data. The main classification theorem (Theorem 5.2) does not rely on the authors' prior paper [16]; that paper is cited only for background on networks in Section 2.3. The proof of Theorem 5.2 attempts to recover the vertex set from the maximal idempotents of E(QΓ) and the relation set from the maximal idempotents of E(QΓ) minus the [A]'s, using Lemma 5.1, and then transfers source/range data via the isomorphism. Whatever the merits of that reconstruction as a correctness matter — and there may indeed be a gap in identifying maximal idempotents with vertex idempotents — it is not circular: the claimed invariant is derived from the semigroup's own idempotent poset, not defined in terms of the network being recovered. All central ingredients are either proved from the presentation or cited to standard external sources, so there is no self-citation chain bearing the load and no constructed quantity that is equal to its input by definition.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants. Its central results rest on the definition of a network (from prior work), the semigroup presentation QGamma, and the confluence of the associated rewriting system, which is proven (with gaps) in the paper. Standard semigroup theory results are used as background.

assumptions (4)

domain assumption Network Gamma=(V,T,s,r) with s(t),r(t) disjoint non-empty subsets of V (Definition 2.7, from [16])
The paper builds on the authors' earlier definition of networks; this is the object class under study.
standard math The semigroup QGamma is defined by the presentation with generators T union T0 union T^{-1} union {0} and relations NR1-NR6 (Definition 3.1)
Semigroup presentations are standard; this is the definition of the object.
ad hoc to paper The reduction system (X+, ->) is noetherian and confluent (Proposition 3.2)
A technical claim proven in the paper, but the proof has gaps; all normal form results depend on it.
standard math Standard results on L*, R*, abundant semigroups, and fundamental inverse semigroups (cited [8,9,10,11,13])
Used as background in Sections 2 and 4.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Network right * abundant semigroups." pith.science (2026). https://pith.science/paper/R34BQC6Q

@misc{pith2026241114897,
  author       = {Pith},
  title        = {Pith review of: Network right * abundant semigroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R34BQC6Q}},
  note         = {Machine review of arXiv:2411.14897}
}

abstract

We introduce the class of network right $*$-abundant semigroups. These are based on networks that extend the notion of a directed graph. This class properly contains the class of graph inverse semigroups. We investigate the structure of network right $*$-abundant semigroups. We show that two network right $*$-abundant semigroups are isomorphic if and only if the underlying networks are isomorphic.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

J Algebra 293 319-334 (2005)

Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J Algebra 293 319-334 (2005)

work page 2005
[2]

A., Pardo, E.: Nonstable K-theory for graph algebras

Ara, P., Moreno, M. A., Pardo, E.: Nonstable K-theory for graph algebras. Algebr Represent Theory, 10 157-178 (2007)

work page 2007
[3]

J., Hall, T

Ash, C. J., Hall, T. E.: Inverse semigroups on graphs. Semigroup F orum 11 140-145 (1975)

work page 1975
[4]

Cambridge University Press, 2022

Bianconi, G.: Higher-Order Networks: An Introduction to Simplicia l Complex. Cambridge University Press, 2022

work page 2022
[5]

M., Cokus, S

Bowers, P. M., Cokus, S. J., Eisenberg, D., et al. Use of Logic Relat ionships to Decipher Protein Network Organization. Science 306 2246-2249 (2004)

work page 2004
[6]

Baird, G.R.: Congruence-free inverse semigroups with zero. J. A ustral. Math. Soc. 20 (Series A) 110-114 (1975)

work page 1975
[7]

V., Otto, F.: String-Rewriting Systems, Springer, 1993

Book, R. V., Otto, F.: String-Rewriting Systems, Springer, 1993

work page 1993
[8]

B.: Adequate semigroups

Fountain, J. B.: Adequate semigroups. Proceeding of the Edinbu rgh Mathematical Society 22(2) 113- 125 (1979)

work page 1979

Show all 17 references

[9]

B.: Abundant semigroups

Fountain, J. B.: Abundant semigroups. Proc Lond Math Soc 44 103-129 (1982)

1982
[10]

B.: Free right h-adequate semigroups

Fountain, J. B.: Free right h-adequate semigroups. Semigroups, theory and applications, Lec ture Notes in Mathematics 1320, 97-120 (1988)

1988
[11]

M.: An Introduction to Semigroup Theory

Howie, J. M.: An Introduction to Semigroup Theory. Academic Pr ess, London (1976)

1976
[12]

Mitsch.: A natural partial order for semigroups

H. Mitsch.: A natural partial order for semigroups. Proceedin gs of the American Mathematical Society 97(3) 384-388 (1980)

1980
[13]

W. D. Munn.: Fundamental inverse semigroups. Quart J Math Ox ford 21 157-170 (1970)

1970
[14]

F.: Une generalisation du monoide bicclique

Nivat, M., Perrot, J. F.: Une generalisation du monoide bicclique. C R Acad Sci Paris, 271 824-827 (1970)

1970
[15]

Paterson, A. L. T.: Graph inverse semigroups, groupoids and t heir C∗ -algebras. J Oper Theory 48 645-662 (2002)

2002
[16]

H., Meng, D

Wang, Y. H., Meng, D. Z.: Quasi-semilattices on networks. Axioms 12 943 (2023) https://doi.org/10.3390/axioms12100943

2023 doi
[17]

Yang, D., Gould, V. A. R.: Free idempotent generated semigroup s over bands and biordered sets with trivial products. International Journal of Algebra and Computa tion 26 473-507 (2016) College of Mathematics and Systems Science, Shandong Unive rsity of Science and Tech- nolog...

2016

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

J Algebra 293 319-334 (2005)

Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J Algebra 293 319-334 (2005)

work page 2005

[2] [2]

A., Pardo, E.: Nonstable K-theory for graph algebras

Ara, P., Moreno, M. A., Pardo, E.: Nonstable K-theory for graph algebras. Algebr Represent Theory, 10 157-178 (2007)

work page 2007

[3] [3]

J., Hall, T

Ash, C. J., Hall, T. E.: Inverse semigroups on graphs. Semigroup F orum 11 140-145 (1975)

work page 1975

[4] [4]

Cambridge University Press, 2022

Bianconi, G.: Higher-Order Networks: An Introduction to Simplicia l Complex. Cambridge University Press, 2022

work page 2022

[5] [5]

M., Cokus, S

Bowers, P. M., Cokus, S. J., Eisenberg, D., et al. Use of Logic Relat ionships to Decipher Protein Network Organization. Science 306 2246-2249 (2004)

work page 2004

[6] [6]

Baird, G.R.: Congruence-free inverse semigroups with zero. J. A ustral. Math. Soc. 20 (Series A) 110-114 (1975)

work page 1975

[7] [7]

V., Otto, F.: String-Rewriting Systems, Springer, 1993

Book, R. V., Otto, F.: String-Rewriting Systems, Springer, 1993

work page 1993

[8] [8]

B.: Adequate semigroups

Fountain, J. B.: Adequate semigroups. Proceeding of the Edinbu rgh Mathematical Society 22(2) 113- 125 (1979)

work page 1979