Small monoids generating varieties with uncountably many subvarieties

Sergey V. Gusev

REVIEW 2 major objections 3 minor 13 references

Small monoids generating varieties with uncountably many subvarieties

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the ten-element Rees quotient monoid M(aabb) generates a variety with uncountably many subvarieties, answering a recent question and yielding a new minimal six-element example.

desk verdict Right open question, right strategy, but the proof of the key transfer step has a real gap; fixable, not fatal to the idea. read the letter →

arxiv 2411.15554 v1 pith:U6HEHW6D submitted 2024-11-23 math.GR

Sergey V. Gusev This is my paper · ORCID

classification math.GR MSC 20M07

keywords monoidvarietyuncountablymanysubvarietiesReesquotientidentitybasisminimalfinitelybased

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 proves that the ten-element Rees quotient monoid M(aabb) — the monoid whose nonzero elements are the subwords of the word aabb — is of type $2^{\aleph_0}$: the variety it generates contains uncountably many distinct subvarieties. This answers a question recently posed in [1] for the analogous monoid M(abba). As a corollary, the author exhibits a six-element monoid that is also of type $2^{\aleph_0}$, is minimal (no smaller monoid has this property), and is finitely based; this is the first minimal example that is finitely based. The proof works by constructing a family of Rees quotient monoids M(W_N), one for each subset N of the positive integers, showing these monoids satisfy a complete identity basis for M(aabb), and showing they generate pairwise distinct varieties. A sympathetic reader should care because it sharpens the boundary between finite monoids that generate small varieties and those that generate wild, uncountable ones.

What carries the argument

The machinery is the Rees quotient construction $M(W)$ for sets of words $W$, together with a depth invariant for letters in words. For each positive integer $n$, the word $w_n$ is built from two known word patterns and has a prescribed depth profile (Lemma 2): the letter $x$ has depth $n+1$, the letters $y_i^{(k)}$ have depth $k$, and the $t_i$, $z_i$ have depths $0$ and $1$. Lemma 1 shows that a substitution into a word with positive depth cannot collapse the first occurrence of a shallower letter. Lemma 3 exploits the unique occurrence of every length-2 subword of $w_k$ to force a contradiction when a substitution maps $w_n$ into $w_k$ with $n \neq k$. This separation, combined with the identity basis $\Sigma$ from [13], is what transfers uncountably many subvarieties into $V(M(aabb))$.

What would settle it

Verify whether the word $w_2$ actually has a unique occurrence of each length-2 subword; if some block repeats, the contradiction in Lemma 3 fails. Alternatively, search for an identity of M(aabb) not derivable from $\Sigma$; such an identity would show the basis is incomplete, and the transfer to $V(M(aabb))$ would be unsupported.

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Extended reading notes

Core claim

The central discovery is that M(aabb) is of type $2^{\aleph_0}$. For each subset $N \subseteq \mathbb{N}$, let $W_N = \{w_n : n \in N\}$ where $w_n$ is a carefully constructed word over an infinite alphabet; the Rees quotient $M(W_N)$ satisfies the identity basis $\Sigma = \{x^3 \approx x^4,\ x^3y \approx yx^3,\ yzx^3 \approx xyxzx,\ xyzxty \approx yxzxty,\ xzytxy \approx xzytyx\}$ that [13] proves for M(aabb), so each $M(W_N)$ lies in the variety generated by M(aabb). Lemma 3 then shows that if $n \notin N$, the monoid $M(W_N)$ satisfies the identity $w_n \approx x^2(w_n)x$, while $M(W_{N \cup \{n\}})$ does not; hence distinct subsets $N$ yield distinct subvarieties. Since there are uncountably many subsets of $\mathbb{N}$, $V(M(aabb))$ has uncountably many subvarieties. The corollary then identifies a six-element monoid $M = \langle a,e \mid ee=e,\ aaa=ae=0,\ eaa=aa\rangle \cup \{1\}$, proves it satisfies these identities, and uses the known classification of small non-finitely based monoids to show $M$ is minimal and finitely based.

Load-bearing premise

The load-bearing premise is that the identity basis $\Sigma$ is complete for M(aabb), and the proof also assumes that every length-2 subword of $w_k$ occurs uniquely.

Editorial extensions

If this is right

M(aabb) joins M(abab) and M(abba) as the only type-$2^{\aleph_0}$ monoids of the form $M(w)$ of order ten or less.
The six-element monoid M is a new minimal example of type $2^{\aleph_0}$ and is finitely based, the first such example.
Every monoid of order five or less generates only countably many subvarieties, so the six-element example is as small as possible.
The identity basis $\Sigma$ transfers all the constructed $M(W_N)$ into $V(M(aabb))$, so the variety generated by a ten-element monoid contains a whole continuum of subvarieties.

Reading between the lines

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

The depth profile of $w_n$ may serve as a general template for separating varieties of Rees quotient monoids beyond the three words aabb, abab, and abba; other finite words with similar depth profiles could yield further type-$2^{\aleph_0}$ examples.
The unproved claim about unique occurrence of length-2 subwords in $w_k$ might be provable by a short combinatorial argument; if it fails for some $k$, Lemma 3 would need repair, but the overall construction might still work with a weaker separation property.
The six-element monoid M, being finitely based, may admit a transparent axiomatization; comparing its lattice of subvarieties with those of the known non-finitely based examples could clarify how the finite-basis property interacts with high cardinality of subvariety lattices.
A testable extension: replace the infinite alphabet with a finite one, such as the letters appearing in the words $w_n$, and ask whether the uncountability phenomenon persists in finite-alphabet Rees quotients.

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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

Short version: this is the right result—M(aabb) being type 2^{aleph_0} was open, and the corollary gives a nice minimal finitely based example—but the proof as written has a real hole in exactly the step that connects the construction to M(aabb). The stress-test note is right: in the verification of the two five-variable Sapir identities, the claim that neither φ(xy) nor φ(yx) is a subword whenever φ(x)≠1 and φ(y)≠1 is false. In w_2, take φ(x)=z_1 and φ(y)=y_1^(2); then φ(xy)=z_1 y_1^(2) appears at positions 6–7. Both letters are multiple, so the repeated-variable intuition does not save it. This is not cosmetic: the argument that M(W_N) ∈ V(M(aabb)) depends on M(W_N) satisfying Σ, and the proof of that satisfaction is incomplete.

What is genuinely good: the construction of w_n is a clean blend of the Jackson and Lee–Zhang patterns, the depth computations in Lemma 2 are straightforward and checked, and the overall strategy is sound. The paper is honest about its external inputs—O. Sapir's identity basis, the Jackson–Sapir reduction, and the Gusev–Vernikov depth lemma—and the self-citation is to a published general tool, not a hidden container of the result. No free parameters or fitted constants.

The other soft spots are minor: the unique-occurrence claim for length-2 subwords of w_k in Lemma 3 is asserted without proof, and there is a typo in the final quotient sentence. The load-bearing gap, though, needs to be fixed before the theorem is established. A likely repair would analyze the full product φ(x)φ(y)φ(z)φ(x)φ(t)φ(y) rather than just the pair φ(xy), or impose additional conditions on the substitution; the present text does not supply that argument.

Who this is for: anyone working on monoid varieties or the lattice of subvarieties, and the corollary is a useful data point. It deserves a serious referee—the question is real, the approach is plausible, and the gap is probably repairable. But I would not accept it in its current form; I would send it back for a substantive revision. My own verdict: skeptical until the gap is closed.

Referee Report

2 major / 3 minor

Summary. The paper studies finite monoids generating varieties with uncountably many subvarieties. It claims that the Rees quotient monoid M(aabb) of order ten is of type 2^{aleph_0}, thereby answering a question of Glasson. The proof uses O. Sapir's identity basis for M(aabb), verifies that a family of monoids M(W_N) satisfies this basis, and uses Lemma 3 to separate the varieties generated by the different M(W_N). A corollary exhibits a six-element monoid that is minimal, finitely based, and of type 2^{aleph_0}.

Significance. If the proof is repaired, the main result would be a notable step: M(aabb) would become the third order-ten Rees quotient known to have uncountably many subvarieties, and Corollary 1 would give the first minimal finitely based monoid of type 2^{aleph_0}. The paper is concise and uses a reasonable toolbox: an external identity basis, the Jackson--Sapir reduction, and the Gusev--Vernikov depth lemma. The proof contains no free parameters, and the straightforward depth computations in Lemma 2 are clean. However, as written, a central verification in the proof of Theorem 1 contains a false inference, so the main claims are not yet established.

major comments (2)

[Proof of Theorem 1] The verification that M(W_N) satisfies Sapir's identity basis Sigma is incomplete. For the identities xyzxty = yxzxty and xzytxy = xzytyx, the paper claims: 'neither phi(xy) nor phi(yx) is a subword of a word in WN because every subword of length 2 of wn consists of the first occurrence of a letter and the last occurrence of a letter in wn.' This inference is false. In w_2, take phi(x)=z_1 and phi(y)=y_1^(2); then phi(xy)=z_1 y_1^(2), which is the length-2 subword at positions 6-7 of w_2, and both letters are multiple in w_2. The stated property of length-2 subwords does not rule out products of two single-letter values occurring as subwords. Consequently the claim that M(W_N) satisfies Sigma is not proven, and the inclusion M(W_N) in V(M(aabb)), which transfers uncountability to M(aabb), is not justified. A complete case analysis is needed.
[Lemma 3] The assertion that every length-2 subword of w_k has a unique occurrence is stated without proof. This property is used to derive (*), which is essential both for the conclusion n <= k and for the lower bound on the non-linear subword in the final contradiction of the lemma. Since Lemma 3 is the mechanism that makes the varieties generated by the M(W_N) pairwise distinct, the uniqueness claim should be verified explicitly or replaced by a precise reference.

minor comments (3)

[Proof of Corollary 1] The sentence 'It is easy to check that M(aabb) satisfies these identities' should be expanded or referenced, since this check is the only step placing M(aabb) in the variety generated by the six-element monoid M.
[Proof of Theorem 1, final paragraph] The phrase 'any monoid of the form M(WN) is a quotient of M(WN)' contains a notational ambiguity: it should distinguish M(W_N) from M(W_N with N = N), since both are written as M(WN).
[Throughout] The text contains several typographical slips, including 'the words wn is' (should be 'are') in the paragraph defining w_n and 'F or' at the start of Lemma 2 (should be 'For').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation relies on external identity bases and a published depth lemma, with no fitted constants or prediction-by-construction.

full rationale

The derivation chain for Theorem 1 is not circular. The uncountability of subvarieties of V(M(aabb)) is transferred from the monoids M(W_N), which are separated by Lemma 3 using the identity wn ≈ x2(wn)x. Lemma 3 invokes Lemma 1, a depth lemma from the author's prior joint work [3], but that lemma is a general, parameter-free statement about substitutions and depths; it does not contain or presuppose the target result that M(aabb) is of type 2^aleph0. The identity basis Σ for M(aabb) is taken from O. Sapir [13, Lemma 3.2(i)], an external source, and Corollary 1 uses an external identity basis for M from Lee and Li [10]. Checking these identities on M(aabb) and then applying Theorem 1 is a standard variety-transfer argument, not a fitted input or a renamed conclusion. There are no free parameters fitted to data, no uniqueness theorem imported from the same authors to forbid alternatives, and no ansatz smuggled in via self-citation. The skeptical concern about the subword argument in the proof of Theorem 1 is a possible proof gap or correctness issue, not a circular reduction: the conclusion does not appear among the assumptions, and the contested inference is not equivalent by construction to the statement being proved. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters and no new unobservable entities. The word patterns w_n are explicit constructions, not fitted quantities. All nontrivial inputs are published lemmas (depth lemma, identity bases, reductions) and one unproved combinatorial assertion about length-2 subwords; these are listed as axioms. The proof does not assume the target result.

assumptions (7)

standard math Standard properties of free monoids, substitutions, and Rees quotients M(W) over an ideal of non-subwords are used throughout.
Definitions in the introduction; no unusual axioms are introduced.
domain assumption Lemma 1 (depth lemma) from Gusev and Vernikov [3, Lemma 3.15] is valid.
Used in Lemma 3 to bound n from the depth of x; if this substitution-depth lemma fails, the proof of Lemma 3 collapses.
domain assumption O. Sapir's identity basis for M(aabb): Sigma = {x^3\approx x^4, x^3y\approx yx^3, yzx^3\approx xyxzx, xyzxty\approx yxzxty, xzytxy\approx xzytyx} is complete.
Invoked in Theorem 1 proof; completeness is essential to conclude M(W_N) belongs to V(M(aabb)).
domain assumption Lee and Li's identity basis for M in Corollary 1 is correct.
Used to prove M(aabb) belongs to V(M), transferring uncountability from M(aabb) to M.
domain assumption Every length-2 subword of w_k has a unique occurrence in w_k.
Asserted in Lemma 3 without proof; needed for claim (*) and the subsequent contradiction.
domain assumption Jackson-Sapir Lemma 5.1 reducing Lemma 3 to the singleton case is valid.
External lemma that allows the proof to consider only substitutions into M(w_k).
domain assumption Background facts: order \le 5 monoids generate at most countably many subvarieties, and A1_2 and B1_2 are the only non-finitely based monoids of order \le 6.
Used to conclude the six-element M is minimal and finitely based.

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Pith. "Pith review of Small monoids generating varieties with uncountably many subvarieties." pith.science (2026). https://pith.science/paper/U6HEHW6D

@misc{pith2026241115554,
  author       = {Pith},
  title        = {Pith review of: Small monoids generating varieties with uncountably many subvarieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6HEHW6D}},
  note         = {Machine review of arXiv:2411.15554}
}

abstract

An algebra that generates a variety with uncountably many subvarieties is said to be of type $2^{\aleph_0}$. We show that the Rees quotient monoid $M(aabb)$ of order ten is of type $2^{\aleph_0}$, thereby affirmatively answering a recent question of Glasson. As a corollary, we exhibit a new example of type $2^{\aleph_0}$ monoid of order six, which turns out to be minimal and the first of its kind that is finitely based.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages

[1]

Semigroup Forum 109, 476–481 (2024)

Glasson, D.: The Rees quotient monoid M(abba) generates a variety with uncountably many subvarieties. Semigroup Forum 109, 476–481 (2024). https://doi.org/10.1007/s00233-024-10463-5

work page doi:10.1007/s00233-024-10463-5 2024
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Limit varieties of monoids satisfying a certain identity

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Jackson, M.: Finiteness properties of varieties and the restriction to ﬁnite algebras. Semigroup Forum 70, 159–187 (2005). https://doi.org/10.1007/s00233-004-0161-x

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work page doi:10.1142/s0218196700000327 2000
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Semigroup Forum 103, 165–190 (2021)

Jackson, M., Zhang, W.T.: From A to B to Z. Semigroup Forum 103, 165–190 (2021). https://doi.org/10.1007/s00233-021-10180-3

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[9]

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Lee, E.W.H.: Advances in the Theory of V arieties of Semig roups. Birkh¨ auser/Springer, Cham (2023). https://doi.org/10.1007/978-3-031-16497-2

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Pith tools

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

[1] [1]

Semigroup Forum 109, 476–481 (2024)

Glasson, D.: The Rees quotient monoid M(abba) generates a variety with uncountably many subvarieties. Semigroup Forum 109, 476–481 (2024). https://doi.org/10.1007/s00233-024-10463-5

work page doi:10.1007/s00233-024-10463-5 2024

[2] [2]

Limit varieties of monoids satisfying a certain identity

Gusev, S.V ., Li, Y .X., Zhang, W.T.: Limit varieties of monoids satisfying a certain identity. Algebra Colloq., to appear. Preprint https://arxiv.org/abs/2107.07120v2

work page Pith review arXiv

[3] [3]

Gusev, S.V ., V ernikov, B.M.: Chain varieties of monoids . Dissert. Math. 534, 1–73 (2018). https://doi.org/10.4064/dm772-2-2018

doi:10.4064/dm772-2-2018 2018

[4] [4]

Semigroup Forum 70, 159–187 (2005)

Jackson, M.: Finiteness properties of varieties and the restriction to ﬁnite algebras. Semigroup Forum 70, 159–187 (2005). https://doi.org/10.1007/s00233-004-0161-x

work page doi:10.1007/s00233-004-0161-x 2005

[5] [5]

Jackson, M., Lee, E.W.H.: Monoid varieties with extreme properties. Trans. Amer. Math. Soc. 370, 4785– 4812 (2018). https://doi.org/10.1090/tran/7091

doi:10.1090/tran/7091 2018

[6] [6]

Jackson, M., McKenzie, R.: Interpreting graph colorabi lity in ﬁnite semigroups. Int. J. Algebra Comput. 16, 119–140 (2006). https://doi.org/10.1142/S0218196706002846

doi:10.1142/s0218196706002846 2006

[7] [7]

Jackson, M., Sapir, O.: Finitely based, ﬁnite sets of wor ds. Int. J. Algebra Comput. 10, 683–708 (2000). https://doi.org/10.1142/S0218196700000327

work page doi:10.1142/s0218196700000327 2000

[8] [8]

Semigroup Forum 103, 165–190 (2021)

Jackson, M., Zhang, W.T.: From A to B to Z. Semigroup Forum 103, 165–190 (2021). https://doi.org/10.1007/s00233-021-10180-3

work page doi:10.1007/s00233-021-10180-3 2021