Answering Five Open Problems Involving Semigroup Conjugacy

Trevor Jack

REVIEW 5 major objections 8 minor 1 cited by

Answering Five Open Problems Involving Semigroup Conjugacy

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Pith's one-line read The paper proves that six standard semigroup conjugacy relations are partition-covering and answers the remaining open problems with explicit counterexamples and an infinite chain of definable conjugacies.

desk verdict Solid partition-covering results and counterexamples in the first half; Section 8's L-completeness claims are not supported as written. read the letter →

arxiv 2411.16284 v1 pith:O47UQXJI submitted 2024-11-25 math.GR

Trevor Jack This is my paper · ORCID

classification math.GR MSC 20M9920E45

keywords semigroupconjugacypartition-coveringnormalbandsinversesemigroupschainsL-completeproblemsone-relationmonoids

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

Semigroup conjugacy is any equivalence relation on a semigroup that agrees with ordinary group conjugacy whenever the semigroup is a group. The paper answers five open problems about such relations, all posed in the literature it responds to. Its central theorem is that each of the six standard relations $\sim_o$, $\sim_c$, $\sim_n$, $\sim_p$, $\sim_p^*$, and $\sim_{tr}$ is partition-covering: for every set $X$ and every partition of $X$, there exists a semigroup on $X$ whose conjugacy classes are exactly the blocks of the partition. The paper also constructs a semigroup embeddable in a group for which $\sim_p$ is not transitive, builds an infinite chain of first-order definable conjugacies, and gives counterexamples for the congruence-quotient and variant-transitivity problems. If these constructions are correct, all five problems are settled as stated.

What carries the argument

The load-bearing objects are the left-zero band construction of Lemma 3.2 and the index-group construction of Theorem 3.3. In the first, each partition block $X_i$ is identified with pairs $\{(i,j) : j \in \alpha_i\}$ for an ordinal $\alpha_i$, and multiplication on $X$ is $(a,b)(c,d) = (\max_P(a,c), b)$ with $\max_P$ comparing the block indices; the result is a normal band in which the relation $\sim_\ell$ (two elements multiply to themselves in both orders) is contained in $\sim_n$, hence in $\sim_p$, $\sim_p^*$, and $\sim_{tr}$, and $\sim_{tr}$-conjugacy forces equality of first coordinates. This sandwich $\sim_\ell \subseteq \sim \subseteq \sim_{tr}$ is what makes all four relations partition-covering. For $\sim_o$ and $\sim_c$, the index set is a commutative group and multiplication $(a,b)(c,d) = (a\cdot c, 0)$ makes the first coordinate behave like group multiplication, so $\sim_o$-conjugacy is equality of first coordinates. The infinite chains are driven by the cycle-chain structure of the full partial bijection semigroup $I_n$: $\sim_{n[k]}$ compares $a^{k+1}a^{-k}$ with $b^{k+1}b^{-k}$, and chains of different lengths separate $\sim_{n[i]}$ from $\sim_{n[j]}$.

What would settle it

Run a brute-force comparison on $I_3$ with $m=1$, checking Algorithm 1's output against the direct existence of an inversive $1$-satisfying pair for every pair of elements; the instance with $a$ a single chain of length 3 and $b$ empty should be rejected, and any mismatch traceable to the arithmetic in line 14 (the update involving $C[i]+a$ rather than $C[i]+t_a$) would refute Lemma 8.1 and Theorem 8.2.

Watch

Extended reading notes

Core claim

The main discovery is Theorem 3.3, which says that $\sim_o$, $\sim_c$, $\sim_n$, $\sim_p$, $\sim_p^*$, and $\sim_{tr}$ are all partition-covering. The proof for the middle four relations uses a normal band built as a semilattice of left-zero subsemigroups: each block of the partition is indexed by an ordinal, elements are pairs $(i,j)$, and multiplication $(a,b)(c,d) = (\max_P(a,c), b)$ makes two elements conjugate exactly when their first coordinates agree. The proof for $\sim_o$ and $\sim_c$ uses a commutative group of indices with multiplication $(a,b)(c,d) = (a\cdot c, 0)$, which again forces conjugacy to coincide with equality of first coordinates. Beyond partition-covering, the paper gives the one-relation monoid $\langle a,b \mid aab = bba\rangle$, which embeds in a group but has nontransitive $\sim_p$, and an infinite chain $\{\sim_{n[k]}\}$ of first-order definable conjugacies on partial bijection semigroups, where $\sim_{n[k]}$ compares $a^{k+1}a^{-k}$ and $b^{k+1}b^{-k}$ under $\sim_n$. It also exhibits an infinite semigroup where $\sim_o$ is a congruence yet the quotient is not cancellative, and an eight-element semigroup where $\sim_p$ fails transitivity while every variant has transitive $\sim_p$.

Load-bearing premise

The paper's claims about $\sim_{p[m]}$ and the L-completeness results rest on Lemma 8.1, which characterizes $\sim_{p[m]}$ by the existence of an 'inversive $m$-satisfying pair,' and on the logarithmic-space algorithm that checks for such pairs; if either the characterization or the algorithm's bookkeeping has a flaw, those claims lose their support.

Editorial extensions

If this is right

For any set and any prescribed partition, one can build a semigroup on that set whose $\sim_o$, $\sim_c$, $\sim_n$, $\sim_p$, $\sim_p^*$, or $\sim_{tr}$ classes realize the partition exactly, so the apparent rigidity of group conjugacy does not carry over to any of these semigroup notions.
The one-relation monoid $\langle a,b \mid aab = bba\rangle$ is embeddable in a group but has nontransitive $\sim_p$, so embeddability in a group does not force transitivity of $\sim_p$.
The relations $\sim_{n[1]} \subsetneq \sim_{n[2]} \subsetneq \cdots$ form an infinite strictly increasing chain of first-order definable conjugacies, and each agrees with group conjugacy when the semigroup is a group.
The decision problems $n[k]$-Conjugacy and $p[m]$-Conjugacy are both L-complete on the full partial bijection semigroup, so the iterated relations do not raise the asymptotic space cost of testing conjugacy.
The eight-element semigroup where every variant is $\sim_p$-transitive while $\sim_p$ itself is not shows that local variant behavior cannot force global transitivity.

Reading between the lines

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

The same ordinal-indexed left-zero construction gives a general template: any equivalence relation on an arbitrary set that can be sandwiched between the left-zero relation and $\sim_{tr}$ on normal bands will be partition-covering, so the family of covering conjugacies is likely much larger than the six listed.
The one-relation monoid example could be adapted to control the number of $\sim_p$ steps explicitly, potentially producing group-embeddable semigroups where $\sim_{p[m]}\neq\sim_{p[m+1]}$, which would extend the chain result to the embeddable setting.
A computational search over small finite semigroups could test whether the infinite example with $\sim_o$ a congruence and noncancellative quotient is necessary, or whether a finite example exists as the paper leaves open.

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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

Jack's note answers five open problems in the semigroup conjugacy program, and the first half is genuinely good. The partition-covering construction in Section 3 is the real contribution: the semilattice-of-left-zero-semigroups trick covers ~n, ~p, ~p*, and ~tr for arbitrary partitions, and the commutative-group construction covers ~o and ~c. That is a clean, self-contained proof of Theorem 3.3. The embeddable monoid counterexample in Section 5 is also convincing—the presentation has acyclic left and right graphs, so Adyan's theorem applies, and the ~p failure is transparent. The non-cancellative quotient example in Section 6 is explicit and checkable.

The soft spots are concentrated in Section 8, and the stress-test note is right. Lemma 8.1's forward induction does not work as written: composing m-satisfying pairs (φ_a,ψ_a) and (φ_c,ψ_c) gives a map whose domain may not align, and the length bound is arithmetically wrong (2m−1 + 2m−1 = 4m−2, not 2m). The converse is garbled, mixing n and m and defining α_i with a typo. Algorithm 1 is not executable: Line 14 uses `a` instead of `ta` and mutates C[i] before shifting it to C[i+1]. So Theorem 8.2 and Proposition 8.3 are unsupported by the text as written. That matters because the paper advertises a new L-complete class, but it is not load-bearing for the five open problems from [3]—those rely on Sections 3–7, which hold up. The self-citation to [7] is appropriate; Lemma 4.4 there is published and distinct.

Bottom line: the core results on partition covering and the counterexamples deserve referee time, but Section 8 needs a rewrite. I'd send it to review with a request to fix or remove the p[m] chain claims and make Algorithm 1 concrete. A serious referee can separate the solid parts from the broken ones.

Referee Report

5 major / 8 minor

Summary. The paper claims to answer five open problems from Araújo, Kinyon, Konieczny, and Malheiro concerning semigroup conjugacy. It proves that the relations ~o, ~c, ~n, ~p, ~p*, and ~tr are partition-covering (Problem One); constructs a semigroup embeddable in a group for which ~p is not transitive (Problem Two); constructs an infinite chain of first-order definable conjugacies ~n[k] (Problem Three); gives an infinite semigroup for which ~o is a congruence and the quotient is not cancellative (Problem Four); and gives a semigroup for which ~p fails transitivity while each variant has transitive ~p (Problem Five). The paper also introduces families ~n[k] and ~p[m] and claims that the associated conjugacy problems are L-complete, with ~p[m] forming a proper chain on the inverse symmetric monoids I_n.

Significance. The partition-covering construction in Section 3 is elegant and, if correct, resolves a natural open question with a clean uniform method. The counterexamples for Problems Two, Four, and Five address questions that have been open in the semigroup literature, and the claimed L-completeness results would be a novel contribution connecting semigroup conjugacy to descriptive complexity. The paper ships several machine-verifiable finite examples and cites prior peer-reviewed work for the base characterizations; these are strengths. However, several load-bearing assertions are currently unproved or sketched, and the Section 8 results on p[m]-Conjugacy are not supported as written, so the contribution is not yet in a publishable state.

major comments (5)

[§5, Corollary 5.2] The proof that aba and bab are not ~p-related is omitted. The monoid relation aab=bba preserves length, but that alone does not rule out all factorizations uv=aba and vu=bab; one must also consider that the only relation cannot be applied to subwords of aba or bab and check the finite possibilities for (u,v). This is a load-bearing step for Problem Two, and the proof needs to be supplied.
[§5, Example 5.3] The property (2), namely that 6·5 and 7·5 are not ~p-related, is asserted without proof. Since this is exactly the failure of transitivity needed for Problem Five, the assertion should be verified, for example by a short exhaustive check of factorizations in the given eight-element semigroup or by citing a GAP computation with the check described.
[§7, Theorems 7.1 and 7.2] The proof of Theorem 7.1 contains an off-by-one error: for the shift map a defined by xa=x+1 on {1,...,j-1}, a^i satisfies x a^i = x+i, not x+i+1. The conclusion that a ~n[j] b and a ⁄~n[i] b for i<j is correct, but the present computation is not. The proof of Theorem 7.2 is also too terse: the alleged logarithmic-space algorithm for n[k]-Conjugacy is not described, and the L-hardness claim 'testing any notion of semigroup conjugacy is L-hard' is imprecise and needs a concrete reduction.
[§8, Lemma 8.1] The induction proving Lemma 8.1 is not valid as written. In the forward direction, the composition (φ_c∘φ_a, ψ_a∘ψ_c) may be undefined because φ_a maps Θ_a^{>2m−1} into Θ_c, not into the required domain Θ_c^{>2m−1} of φ_c. The length estimate is also arithmetically wrong: 2m−1+2m−1 is 4m−2, not 2m, and it does not yield the required 2m+1 bound for an (m+1)-satisfying pair. In the converse, the construction mixes n and m (e.g., 'Θ >2n−1 a' and 'Θ 2n−1 b') and the definition of α_i is garbled. Since Lemma 8.1 underpins Theorem 8.2 and Proposition 8.3, those results are currently unsupported.
[§8, Algorithm 1 and Theorem 8.2] Algorithm 1 is not executable as written: Line 14 uses the input element a where the integer ta is intended, and Line 16 shifts C[i] into C[i+1] after C[i] has already been updated, which can overwrite data needed in later iterations. The correctness proof does not resolve these problems or explain the intended behavior. Consequently, the claim that p[m]-Conjugacy can be decided in logarithmic space (Theorem 8.2) is not established.

minor comments (8)

[Abstract] There is a typo: 'equal s group conjugacy' should read 'equals group conjugacy'.
[§3, Lemma 3.2] The notation max P(i,j) is confusing; writing max_P(i,j) or defining it inline would improve readability.
[§3, Theorem 3.3] In the third construction for the singleton partition, it would help to state explicitly that the left-zero semigroup makes ~c the universal relation because every element acts as an identity for the others in the required sense.
[§4, Example 4.2] The claim that [3,2,4,4] and [4,2,3,3] are not ~n-related is not justified; a brief kernel/image argument or a note that this was checked in GAP would be useful.
[§6, Example 6.1] The proof that the listed sets are exactly the ~o-classes is sketched; the argument that elements of the form b^i a and b^j a for i≠j are unrelated is not explicitly given and should be included.
[§7, Theorem 7.2] The adaptation of the algorithms from [7] is described only as 'simply ignoring chains of length less than k'; a more precise description of the modified algorithm and its space usage is needed.
[§8] The notation ~p[m] for the relation and p[m]-Conjugacy for the decision problem is potentially confusing; a sentence explicitly distinguishing the two would help.
[Throughout] The manuscript contains numerous small typos (e.g., 'convenction', 'cancelative' for 'cancellative', 'satsifying') and should be carefully proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central constructions are self-contained, and the author's prior work [7] is used only as ordinary published support, not as a self-referential premise.

full rationale

The main derivations in the paper do not reduce to their own inputs. Section 3 explicitly constructs semigroups realizing arbitrary prescribed partitions, and verifies by direct multiplication that the six conjugacies coincide with the block relation; the proof is self-contained and does not assume the partition-covering conclusion. Sections 4-6 answer the remaining open problems with explicit examples, external theorems such as Adyan's embeddability criterion, and small-semigroup tables, none of which are fitted parameters or renamed versions of the target claims. Section 7 defines the new relations ~n[k] in terms of the known ~n cycle-chain characterization and proves the chain theorem algebraically using external results [2,8,9,10]; the definition is not a restatement of the conclusion. Section 8 extends the author's earlier characterization [7, Lemma 4.4] to ~p[m] by induction. This is reliance on a prior published result with distinct content, not circularity: the base case is not the theorem being proved, and the present Lemma 8.1 adds a new induction over m. The L-completeness proof also cites [7] for lower bounds and for the ~p* logspace test; these are independent published results, not self-referential assumptions of the present claims. The apparent typographical and correctness issues in Algorithm 1 and in the induction estimates are legitimate correctness concerns, but they do not constitute a circular reduction: no equation is defined in terms of the target relation, and no fitted value is later announced as a prediction. Under the quoted-reduction standard required for a circularity finding, no specific circular step can be exhibited, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central claims rely on standard set-theoretic choice, on the cited chain of inclusions and characterizations, and on Adyan's embeddability theorem. No free parameters are introduced. The families ~n[k] and ~p[m] are explicit definitions with proven properties, not unconstrained postulates.

assumptions (6)

standard math Axiom of choice: every set can be well-ordered.
Lemma 3.2 uses Von Neumann ordinals and well-ordering of each Xi to index elements of the constructed semigroup.
domain assumption Known inclusion chain ~n ⊊ ~p ⊊ ~p* ⊊ ~tr ⊊ ~o and ~n ⊊ ~c ⊊ ~o (Eq. 2.1 of [3]).
Used in Lemma 3.2 and Theorem 3.3 to infer that the constructed left-zero relation lies inside the target conjugacies.
domain assumption Adyan's theorem: a finitely presented monoid embeds in a group if neither its left nor its right graph has cycles.
Used in Theorem 5.1 and Corollary 5.2 to prove that the one-relation monoid S is embeddable in a group.
domain assumption Cycle-chain characterization of ~n for inverse semigroups ([8, Thm 2.6], [2, Thm 2.10]).
Used in Theorems 7.1 and 7.2 to compute classes of ~n[k] and to establish L-membership.
domain assumption Characterization of ~p via inversive 1-satisfying pairs ([7, Lemma 4.4]) and L-completeness of group conjugacy ([7, Thm 4.6]).
Used as the base case and hardness argument for Lemma 8.1 and Theorem 8.2; this is the author's own prior published work.
standard math For every cardinal k, there exists a commutative group of cardinality k.
Used in Theorem 3.3 to construct the index group (I, ·) for the ~o and ~c partition-covering construction.

invented entities (2)

The family of conjugacy relations ~n[k] for k in N independent evidence
purpose: Forms an infinite chain of first-order definable semigroup conjugacies, answering the chain half of Problem 6.21.
Explicitly defined as a ~n[k] b iff a^(k+1) a^(-k) ~n b^(k+1) b^(-k); Theorem 7.1 proves strict inclusions and Theorem 7.2 proves L-completeness.
The family of relations ~p[m] for m in N independent evidence
purpose: Provides an infinite chain of relations extending ~p that equal group conjugacy on groups, and yields new L-complete problems.
Defined by iterating the ~p relation; Lemma 8.1 and Proposition 8.3 establish distinctness and L-completeness.

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Pith. "Pith review of Answering Five Open Problems Involving Semigroup Conjugacy." pith.science (2026). https://pith.science/paper/O47UQXJI

@misc{pith2026241116284,
  author       = {Pith},
  title        = {Pith review of: Answering Five Open Problems Involving Semigroup Conjugacy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O47UQXJI}},
  note         = {Machine review of arXiv:2411.16284}
}

read the original abstract

A semigroup conjugacy is an equivalence relation that equals group conjugacy when the semigroup is a group. In this note, we answer five open problems related to semigroup conjugacy. (Problem One) We say a conjugacy ~ is partition-covering if for every set X and every partition of the set, there exists a semigroup with universe X such that the partition gives the ~-conjugacy classes of the semigroup. We prove that six well-studied conjugacy relations -- ~o, ~c, ~n, ~p, ~p*, and ~tr -- are all partition-covering. (Problem Two) For two semigroup elements a and b in S, we say a ~p b if there exists u and v in S such that a=uv and b=vu. We give an example of a semigroup that is embeddable in a group for which ~p is not transitive. (Problem Three) We construct an infinite chain of first-order definable semigroup conjugacies. (Problem Four) We construct a semigroup for which ~o is a congruence and S\~o is not cancellative. (Problem Five) We construct a semigroup for which ~p is not transitive while, for each of the semigroup's variants, ~p is transitive.

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Works this paper leans on

14 extracted references · 12 canonical work pages · cited by 1 Pith paper

[3]

Ara´ ujo, M

J. Ara´ ujo, M. Kinyon, J. Konieczny, and A. Malheiro. Fou r notions of conjugacy for ab- stract semigroups. Proceedings of the Royal Society of Edinburgh: Section A Mat hematics, 147(6):1169–1214, 2017. doi:10.1017/S0308210517000099

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Pith tools

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

[1] [3]

Ara´ ujo, M

J. Ara´ ujo, M. Kinyon, J. Konieczny, and A. Malheiro. Fou r notions of conjugacy for ab- stract semigroups. Proceedings of the Royal Society of Edinburgh: Section A Mat hematics, 147(6):1169–1214, 2017. doi:10.1017/S0308210517000099

work page doi:10.1017/s0308210517000099 2017

[2] [7]

T. Jack. On the complexity of inverse semigroup conjugac y. Semigroup Forum 106, 618–632 (2023). https://doi.org/10.1007/s00233-023-10349-y

doi:10.1007/s00233-023-10349-y 2023

[3] [1]

S. I. Adyan. On the embeddability of semigroups in groups . Soviet Math. Dokl. , 1:819-821, 1960

work page 1960

[4] [2]

Ara´ ujo, M

J. Ara´ ujo, M. Kinyon, and J. Konieczny. Conjugacy in in- verse semigroups. Journal of Algebra , 533:142–173, 2019. URL:https://www.sciencedirect.com/science/article/pii/S0021869319302765 . doi:10.1016/j.jalgebra.2019.05.022

doi:10.1016/j.jalgebra.2019.05.022 2019

[5] [4]

Ara´ ujo, J

J. Ara´ ujo, J. Konieczny, and A. Malheiro. Conjugation i n semigroups. J. Algebra, 403:93–134,

work page

[6] [5]

Distler and J

A. Distler and J. Mitchell. Smallsemi, a library of small semigroups, Version 0.6.13. URL: https://gap-packages.github.io/smallsemi/, , Feb 2022. GAP package. ANSWERING FIVE OPEN PROBLEMS INVOL VING SEMIGROUP CONJUGAC Y 15

work page 2022

[7] [6]

Izhakian, J

Z. Izhakian, J. Rhodes, and B. Steinberg. Representatio n theory of ﬁnite semigroups over semirings. J. Algebra 336:139–157, 2011. doi:10.1016/j.jalgebra.2011.02.048

work page doi:10.1016/j.jalgebra.2011.02.048 2011

[8] [8]

Konieczny

J. Konieczny. A new deﬁnition of conjugacy for semigroup s. Journal of Algebra and Its Appli- cations, 17(02):1850032, 2018. https://doi.org/10.1142/S0219498818500329

work page doi:10.1142/s0219498818500329 2018