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Transformation Semigroups Which Are Disjoint Union of Symmetric Groups

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.15081 v1 pith:PXJWYSDZ submitted 2024-11-22 math.RA

Utsithon Chaichompoo , Kritsada Sangkhanan This is my paper · ORCID
classification math.RA MSC 20M1720M1920M20
keywords transformationsemigroupequivalencerelationrightgrouprankmaximalsubsemigroupkernelidempotentsymmetric
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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)).

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 (3)
  1. [§4, Corollary 4.4] 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.
  2. [§6, Example 6] 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.
  3. [§2 (Corollary 2.2, Theorem 2.3, Lemma 2.9; used in Proposition 3.4)] 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.
minor comments (4)
  1. [Abstract] 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 '∅'.
  2. [§6, Example 6] 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.
  3. [§5] The phrase 'symmetric group of order n' should be 'symmetric group of degree n', since the order of S_n is n!, not n.
  4. [§4, Theorem 4.3] 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.

Circularity Check

1 steps flagged · score 4.0 of 10

The new isomorphism, rank, and maximal-subsemigroup results are non-circular conditional theorems, but their structural premise—that Q_E*(X) is a right group that is a union of symmetric groups—is imported from the corresponding author's own prior preprint [9], making the self-citation load-bearing.

  1. self citation load bearing [Section 2, Corollary 2.2 / Theorem 2.3 / Remark 2.4 / Lemma 2.9, applied in Section 3 (Proposition 3.4), Section 4 (Corollary 4.4), and Section 5.]
    "Corollary 2.2 ([9, Corollary 3.8]). For each α ∈ QE∗(X), Hα = {β ∈ QE∗(X) : Xα = Xβ} forms a subgroup of QE∗(X). Theorem 2.3 ([9, Theorem 3.9]). QE∗(X) is a union of symmetric groups. ... Lemma 2.9 ([9, Lemma 4.1]). α ∈ QE∗(X) is an idempotent if and only if Aα ⊆ A for all A ∈ X/E."

    Proposition 3.4 states QE∗(X) is isomorphic to S_{X/E} × E(QE∗(X)), citing [9] for the fact that each H-class is a symmetric group on a cross-section and Lemma 2.9 for the count of idempotents. Every subsequent main result (Theorem 3.5, Corollary 4.4, and the Section 5 maximal-subsemigroup count) is derived from this product decomposition. The decomposition is not proved in the paper; its content comes from [9], which is the corresponding author's own earlier arXiv preprint. Thus the central structure is carried by a self-citation. The paper's own rank and isomorphism arguments are logical consequences of that assumed structure, so the circularity is limited to this load-bearing import rather than being a definitional identity.

full rationale

I walked the derivation chain from the definitions of T_E*(X) and Q_E*(X) through Sections 2–5. The semigroup Q_E*(X) is defined directly from the equivalence relation, and the paper's new contributions—the isomorphism criterion (Theorem 3.5), the rank formula (Corollary 4.4, modulo the known correctness issue for infinite X), and the maximal-subsemigroup count—are proved from the product decomposition Q_E*(X) ≅ S_{X/E} × E(Q_E*(X)). That decomposition is not assumed in the definition; it is assembled from Lemma 3.3 and from structural facts stated as Corollary 2.2, Theorem 2.3, Remark 2.4, and Lemma 2.9, all attributed to [9], the corresponding author's prior preprint. No step in this paper says, for instance, 'rank(Q_E*(X)) = m because |E(Q_E*(X))| = m,' nor does any theorem reuse its own conclusion as a hypothesis; the rank and maximal-subsemigroup arguments are genuine derivations from the stated right-group structure. Therefore there is no definitional or fitted-input circularity. The significant caveat is that the structural premise itself is not independently established in the present paper: it is imported via self-citation, and all three main theorems would collapse if the [9] results were unsound. Under the rubric this is load-bearing self-citation rather than a reduction-to-inputs, so the score is 4 rather than 6–10. I also note the reviewer's correctness concern (Corollary 4.4 is stated for arbitrary nonempty X but uses the false claim that every infinite symmetric group has rank 2); that affects correctness, not circularity, and does not change this score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The main external dependency is the prior preprint [9], which supplies the right-group structure of Q_E*(X); the new results are derived from that structure plus standard semigroup theory.

assumptions (3)
  • standard math Standard semigroup theory: Green's relations, right group structure theorem (right group is G times E with E right zero), R-class characterization in T(X) by kernel.
    Used in Theorem 2.6, Proposition 3.4, and Theorem 4.3; all standard and cited to [1].
  • domain assumption X finite and E nontrivial in the rank and maximal-subsemigroup results.
    Explicitly stated in Corollary 4.4 and Section 5; ensures symmetric group rank facts and finiteness of maximal subgroups.
  • ad hoc to paper Structural results from [9] are correct: Q_E*(X) is the kernel of Reg(T), a right group, H_alpha is isomorphic to S_{Xalpha}, and idempotents satisfy Aalpha subset of A.
    Load-bearing and not reproved here; cited to the corresponding author's own arXiv preprint (Corollary 2.2, Theorem 2.3, Lemma 2.9).

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Pith. "Pith review of Transformation Semigroups Which Are Disjoint Union of Symmetric Groups." pith.science (2026). https://pith.science/paper/PXJWYSDZ

@misc{pith2026241115081,
  author       = {Pith},
  title        = {Pith review of: Transformation Semigroups Which Are Disjoint Union of Symmetric Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PXJWYSDZ}},
  note         = {Machine review of arXiv:2411.15081}
}
abstract

Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$ T_{E^*}(X)=\{\alpha\in T(X):\text{for all}\ x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}. $$ We have the regular part of $T_{E^*}(X)$, denoted by $\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by $$ Q_{E^*}(X)=\{\alpha\in T_{E^*}(X):|A\alpha|=1\ \text{and}\ A\cap X\alpha\neq\emptyset\ \text{for all}\ A\in X/E\}. $$ Then we can prove that this subsemigroup is the (unique) minimal ideal of $\mathrm{Reg}(T)$ which is called the kernel of $\mathrm{Reg}(T)$. In this paper, we will compute the rank of $Q_{E^*}(X)$ when $X$ is finite and prove an isomorphism theorem. Finally, we describe and count all maximal subsemigroups of $Q_{E^*}(X)$ where $X$ is a finite set.

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Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [9]

    The regular part of transformation semigroups that preserve double direction equivalence relation

    K. Sangkhanan. The regular part of transformation semig roups that preserve dou- ble direction equivalence relation. arXiv e-prints , page arXiv:2306.08932, June 2023

    work page Pith review arXiv 2023

  2. [1]

    A. H. Clifford and G. B. Preston. The Algebraic Theory of Semigroups, V olume I. American Mathematical Society, 1961

    work page 1961

  3. [2]

    L. Deng, J. Zeng, and B. Xu. Green’s relations and regular ity for semigroups of transformations that preserve double direction equivalen ce. Semigroup F orum, 80(3):416–425, 2010

    work page 2010

  4. [3]

    D. S. Dummit and R. M. Foote. Abstract Algebra. Wiley, 3rd edition, 2004

    work page 2004

  5. [4]

    J. C. Harden. Direct and Semidirect Products of Semigroups . Master’s thesis, 1949

    work page 1949

  6. [5]

    J. M. Howie. Fundamentals of semigroup theory . London Mathematical Society monographs. Clarendon ; Oxford University Press, Oxford Ne w Y ork, 1995

    work page 1995

  7. [6]

    K. D. Magill. A survey of semigroups of continuous selfma ps. Semigroup F orum, 11(1):189–282, 1975

    work page 1975

  8. [7]

    Mitchell and W

    J. Mitchell and W . A. Wilson. Number of maximal subgroups of the symmetric group S_n. The On-Line Encyclopedia of Integer Sequences , 2017

    work page 2017

Show all 12 references
  1. [8]

    H. Pei. Regularity and Green’s relations for semigroups of transformations that preserve an equivalence. Communications in Algebra, 33(1):109–118, 2005

    2005

  2. [10]

    L. Sun, H. Pei, and Z. Cheng. Naturally ordered transfor mation semigroups preserving an equivalence. Bulletin of the Australian Mathematical Society , 78(1):117–128, 2008. 12

    2008

  3. [11]

    Sun and J

    L. Sun and J. Sun. A partial order on transformation semi groups that preserve double direction equivalence relation. Journal of Algebra and Its Applications , 12(8):1350041, 2013

    2013

  4. [12]

    W . A. Wilson. Computational techniques in finite semigroup theory . Thesis, 2018. KRITSADA SANGKHANAN, Department of Mathematics, Faculty o f Science, Chiang Mai University, Chiang Mai, 50200, Thailand; e-mail : kritsada.s@cmu.ac.th UTSITHON CHAICHOMPOO, Doctor of Philosophy P...

    2018

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