Counting finite semirings

J. D. Mitchell; J. Edwards; P. Ragavan

REVIEW 2 major objections 4 minor 1 cited by

Counting finite semirings

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Pith's one-line read This paper establishes exact enumeration counts for finite semirings of order at most 6, using a double-coset reduction that also yields selected classes up to order 8.

desk verdict Solid enumeration data; the n=6 counts are new and likely correct, but the paper needs to state explicitly how the Smallsemi anti-isomorphism convention is handled. read the letter →

arxiv 2507.03709 v3 pith:5TIYC4NX submitted 2025-07-04 math.RA

J. Edwards , J. D. Mitchell , P. Ragavan This is my paper · ORCID

classification math.RA MSC 16Y6020M1005A15

keywords semiringsfiniteenumerationisomorphismclassesadditivelyidempotentcommutativedoublecosetssmallsemigrouplibrary

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 establishes exact counts of finite semirings of small order: up to isomorphism and up to isomorphism-or-anti-isomorphism for every order $n\leq 6$, with selected special classes carried to $n=7$ or $n=8$. The headline numbers are $7{,}571{,}579$ semirings of order $6$ up to isomorphism and $4{,}102{,}358$ up to isomorphism or anti-isomorphism. The enumeration rests on a double-coset criterion that tests distributivity on one representative of each double coset rather than on every relabelling, together with an existing complete library of small semigroups. A sympathetic reader would care because exact counts for unconstrained algebraic structures at these orders were largely missing, and because the paper resolves a discrepancy over the number of order-$5$ semirings.

What carries the argument

The engine is Theorem 1.3, the double-coset criterion: for a commutative additive semigroup $(S,+)$ and semigroup $(S,\times)$, the relabelled multiplications $\times^\sigma$ and $\times^\tau$ are semigroup-isomorphic through a bijection in $\operatorname{Aut}(S,+)$ iff $\sigma$ and $\tau$ belong to the same double coset in $\operatorname{Aut}(S,\times)\setminus\operatorname{Sym}(S)/\operatorname{Aut}(S,+)$. Corollaries 1.4 and 1.6 convert this into an enumeration rule: to count semirings up to isomorphism, it suffices to walk over all commutative semigroups $(S,+)$ and all semigroups $(S,\times)$ from the library, and to test distributivity of $+$ against $\times^\sigma$ for one representative $\sigma$ of every double coset. This replaces the naive process of testing every permutation with a much smaller representative set, and the representatives are computed from the automorphism groups of the two semigroups, which are themselves obtained by reducing automorphism-group computation to graph automorphism.

What would settle it

Write an independent program that, on a four-element set, enumerates all pairs of binary operations satisfying commutativity, associativity, and distributivity and then quotients by relabelling; if the number of resulting isomorphism classes is not $2{,}341$, or the additively idempotent subclass is not $866$, the tables in this paper are wrong.

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Extended reading notes

Core claim

For a fixed commutative additive semigroup $(S,+)$ and a fixed semigroup $(S,\times)$, every semiring on $S$ whose additive part is $+$ and whose multiplication is a relabelling of $\times$ corresponds to a permutation $\sigma\in\operatorname{Sym}(S)$. The paper proves that two such relabelled multiplications $\times^\sigma$ and $\times^\tau$ give isomorphic semirings exactly when $\sigma$ and $\tau$ lie in the same double coset $\operatorname{Aut}(S,\times)\setminus\operatorname{Sym}(S)/\operatorname{Aut}(S,+)$, with an analogous statement, using $\operatorname{Aut}^*(S,\times)$, governing isomorphism or anti-isomorphism. The paper's central claim is that, by checking distributivity for one representative of each double coset and drawing the candidate semigroups from a precomputed library of all semigroups of order at most $8$, the totals in Tables 1--4 are the exact numbers of finite semirings up to the stated equivalences. These include $7{,}571{,}579$ semirings of order $6$ up to isomorphism, $4{,}102{,}358$ up to isomorphism or anti-isomorphism, and the confirmed value of $866$ additively idempotent semirings of order $4$ (semirings in which $x+x=x$ for every element).

Load-bearing premise

The completeness and correctness of the precomputed library of semigroups of order at most 8 is assumed, and a missing or mislabelled isomorphism class there would make every reported semiring count too small.

Editorial extensions

If this is right

The order-6 counts, $7{,}571{,}579$ up to isomorphism and $4{,}102{,}358$ up to isomorphism or anti-isomorphism, give future enumeration algorithms a concrete numerical benchmark to match.
The corrected order-5 total of $57{,}427$ replaces the conflicting figure of $57{,}443$ that had come from another enumeration, so any database or theorem relying on the larger number needs to be rechecked.
The verification of the order-4 additively idempotent count ($866$) settles the numerical question raised in the finite-basis study that prompted this paper.
The same double-coset reduction applies to any subclass of semirings, which is why the tables also cover commutative, with-zero, with-one, and additively idempotent variants, in some cases up to order $8$.

Reading between the lines

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

A likely next step is to compute the missing order-7 total for unconstrained semirings up to isomorphism; the paper already lists the order-7 total up to isomorphism-or-anti-isomorphism ($48{,}152{,}448{,}707$), so the missing figure is plausibly within reach at comparable computational cost.
Because the cost of the method scales with the number of automorphism-group double cosets rather than with $n!$ relabellings, the same program may extend to order $8$ for classes whose semigroups have small automorphism groups; that scaling statement is an editorial guess, not a claim in the paper.
The discrepancy the paper found in an independent library suggests that other generated collections of 'semirings' should be screened for violation of distributivity before being used in downstream work, since the offending object can be an invalid multiplication table rather than a count.
The criterion treats the two operations asymmetrically only through their automorphism groups, so the same double-coset scheme could count other two-operation algebras with one commutative and one associative operation whenever an inventory of the second operation exists.

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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

This is a good, honest enumeration paper. The genuinely new content is the n=6 row of Table 1 and the up-to-anti-isomorphism counts, which to my knowledge have not appeared before. The method is elementary but effective: use Corollaries 1.4 and 1.6 to reduce the problem to iterating over double cosets for each candidate additive and multiplicative semigroup. Theorem 1.3 is correct; the proof has a small typographic slip in the (=>) direction, but the intended argument reconstructs cleanly. The external checks are reassuring: n<=5 up-to-isomorphism counts match MathStructures and the 866 ai-semirings of order 4 from Ren et al. The code is public, and the discrepancy with alg is explained concretely with a reference to a GitHub issue.

The soft spot is the one the stress-test flagged. The paper says the semigroups from Smallsemi are available "up to isomorphism and anti-isomorphism" without clarifying what that means for the candidate list. For the central "up to isomorphism" tables, you need every isomorphism class of multiplicative semigroups, not merely one representative from each anti-isomorphism pair. If Smallsemi stores only one orientation per anti-pair (which is its usual convention), then the algorithm as written would miss the semirings whose multiplicative part is the missing opposite, and the up-to-isomorphism counts would be too low. The fact that the n<=5 counts match the literature suggests the implementation actually handles this correctly--perhaps the code adds the opposite of each non-self-dual semigroup, or Smallsemi for orders <=7 stores full isomorphism-class representatives. But the paper never says which. That is an exposition gap, not necessarily a computational error, but it is load-bearing for the paper's main claim. The authors should add one sentence describing exactly what list of semigroups is iterated over and how anti-isomorphic pairs are treated.

The other limitations are minor: the n=6 and n=7 numbers are not independently verified, so confidence is moderate; and the completeness of Smallsemi is assumed without discussion. Neither is fatal for a data paper of this sort.

This is a paper for semigroup and universal algebra people who want exact counts for testing conjectures, not for readers seeking new theory. It deserves a serious referee--with a request that the candidate-list issue be clarified. I would not cite it in my own work, but I would point colleagues to it.

Referee Report

2 major / 4 minor

Summary. This paper presents a computational enumeration of finite semirings of small order. The authors prove a double-coset criterion (Theorem 1.3) that reduces the enumeration of semirings with fixed additive and multiplicative semigroups to testing distributivity for one representative of each double coset Aut(S,×)\Sym(S)/Aut(S,+) (and the analogous criterion with Aut* for anti-isomorphism). Using the GAP packages Smallsemi and Semigroups, they produce tables of counts for n≤6 (and some classes up to n=8), including the verification that there are 866 ai-semirings of order 4, and they report a discrepancy with the alg package for order 5, which they attribute to 16 invalid multiplication tables in alg.

Significance. If the counts are correct, they are the first exact counts for semirings of order 6 and several classes of order 7 and 8, and the method is an elegant, parameter-free reduction that will be useful for further enumeration. The paper ships reproducible code in the GAP package Semirings, and the double-coset criterion is proved in the paper. The external agreement for orders up to 5 (with MathStructures and Ren et al.) provides strong evidence of correctness. The main risk is the reliance on the Smallsemi library for the semigroup representative lists, and the lack of an independent check for n=6 and n=7.

major comments (2)

[Section 2, algorithm description] The text says that 'it suffices to consider every commutative semigroup (S,+) and semigroup (S,×)' and then notes that Smallsemi contains semigroups 'up to isomorphism and anti-isomorphism.' For the columns labeled 'up to isomorphism' in Tables 1-3, the multiplicative semigroups must be enumerated up to isomorphism, not merely up to anti-isomorphism. If Smallsemi stores only one representative of each anti-isomorphism pair, then for every semigroup S that is not anti-isomorphic to itself, the opposite semigroup S^op is absent from the iteration, and every semiring with multiplicative part S^op is omitted from the up-to-isomorphism counts. The disparity between the n=2 entries (10 versus 9) shows that the implementation must in fact use both orientations, or a list up to isomorphism, but the paper does not state this. Please state explicitly which representative list is used for each column and how the opposite orientation is handled, since the n=6 and n=7 counts have no external cross-check.
[Section 2, last paragraph before Table 1] The paper assumes the completeness and correctness of the Smallsemi library for semigroups of order up to 8. The cross-checks against alg and MathStructures for n≤5 are valuable, but for n=6 and n=7 there is no independent verification, and any missing isomorphism class in Smallsemi would directly change the reported counts. I do not regard this as a flaw in the method, but the paper should explicitly acknowledge this dependence, state the exact version and options used (the reference gives Version 0.7.2), and ideally provide additional validation, such as an independent generation of semigroups of order 6 up to isomorphism from a different source or a consistency check via Burnside's lemma on the number of semirings.

minor comments (4)

[Section 1, proof of Theorem 1.3] In the reverse direction, the displayed chain of equalities contains a stray fragment 'τ α^{-1} ∈ Aut(S,×)' in the middle of the derivation; this should be re-typeset so that the reader can follow the argument without ambiguity.
[Section 2] The phrase 'up to isomorphism and anti-isomorphism' is ambiguous; please use 'up to isomorphism or anti-isomorphism' or explicitly spell out the equivalence relation being used for the semigroup representatives.
[Table 1 caption] The table uses 'with 1' without an explicit definition; please state in the caption or the introduction that this means a multiplicative identity element, matching the earlier definition of 'with 0'.
[Section 2, reproducibility] The paper mentions that the longest computation took approximately 2400 CPU hours; for reproducibility, consider reporting the total CPU time and the exact versions of GAP, Smallsemi, and Semigroups used, and provide a small sample script that reproduces one of the table entries.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the semiring counts are the output of an independent enumeration whose only external input is a prior semigroup library; no fitted parameter or self-referential equation appears.

full rationale

The derivation chain is self-contained with respect to the claimed semiring counts. Section 2 explains that the computation enumerates, for each commutative additive semigroup (S,+) and multiplicative semigroup (S,×) from Smallsemi, one representative σ of each double coset Aut(S,×)\Sym(S)/Aut(S,+), and then checks distributivity via Corollaries 1.4 and 1.6. The double-coset theorems are proved in the paper, not imported by citation. No parameter is fitted to any semiring count, and no table entry is used to produce another table entry. The central assumption is the completeness of Smallsemi, a prior library of semigroups up to order 8; this is an input to the computation, not a consequence of it. Although Smallsemi is coauthored by J. D. Mitchell, its content is an independent semigroup enumeration whose assumptions do not include any semiring counts, so under the stated rules it is real evidence rather than circular self-citation. The paper also checks its results against independent sources: the 866 ai-semirings of order 4 from Ren et al. [46], the MathStructures tables [8–10], and alg [2], with the one discrepancy explicitly traced to an apparent distributivity bug in alg [3]. The sentence in Section 2 that semigroups of order at most 8 are available in Smallsemi 'up to isomorphism and anti-isomorphism' raises a possible completeness issue for the up-to-isomorphism columns if both orientations were not expanded, but that is a correctness concern about the input library, not a circular reduction of the paper's own argument. There is no equation, definition, or fitted parameter that identifies a claimed output with an input by construction, so the appropriate circularity score is 0.

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

No free parameters are fitted to data; the counts are exhaustive computational results. No new objects are postulated. The central claim rests entirely on the correctness of the underlying semigroup library and the GAP implementation.

assumptions (3)

domain assumption The Smallsemi library contains exactly one representative of each isomorphism or anti-isomorphism class of semigroups of order at most 8.
The enumeration in Section 2 iterates over this library; any missing isomorphism class would make every count in Tables 1 to 4 too small. The paper cites the package but does not prove completeness or correctness.
domain assumption Automorphism groups of semigroups of order at most 8 are computed exactly by reduction to graph automorphism (Miller, bliss), and GAP's double coset routine returns a complete set of representatives.
The double coset reduction in Theorem 1.3 requires exact Aut(S,+) and Aut(S,×); the authors rely on the Semigroups GAP package and bliss for these computations in Section 2.
domain assumption Distributivity of + and ×σ for every representative σ is checked correctly by the Semirings GAP package.
The counts are the result of these checks; no formal proof certificates are supplied.

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Cite this review

Pith. "Pith review of Counting finite semirings." pith.science (2026). https://pith.science/paper/5TIYC4NX

@misc{pith2026250703709,
  author       = {Pith},
  title        = {Pith review of: Counting finite semirings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5TIYC4NX}},
  note         = {Machine review of arXiv:2507.03709}
}

abstract

In this short note we count the finite semirings up to isomorphism, and up to isomorphism or anti-isomorphism for some small values of $n$; for which we utilise the existing library of small semigroups in the GAP package Smallsemi.

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The finite basis problem for matrix semirings $\mathbf{M}_n(S_7)$
math.RA 2026-06 accept novelty 6.5 of 10

Every variety between Sc(abc) and Mn(S7) is nonfinitely based, so Mn(S7) itself is nonfinitely based for all n≥2 and the interval from V(S7) is infinite.

Reference graph

Works this paper leans on

48 extracted references · 44 canonical work pages · cited by 1 Pith paper

[1]

Aichinger et al.SONATA, System of nearrings and their applications, Version 2.9.6

E. Aichinger et al.SONATA, System of nearrings and their applications, Version 2.9.6. Refereed GAP package. Dec. 2022.url:https://gap-packages.github.io/sonata/

work page 2022
[2]

Bauer.alg: Algebraic library for constructive mathematics.https://github.com/andrejbauer/alg

A. Bauer.alg: Algebraic library for constructive mathematics.https://github.com/andrejbauer/alg. Accessed: 2025-07-03. 2019

work page 2025
[3]

Bauer.Issue #16: Incorrect results.https://github.com/andrejbauer/alg/issues/16

A. Bauer.Issue #16: Incorrect results.https://github.com/andrejbauer/alg/issues/16. GitHub issue, opened Jul 4, 2025. 2025

work page 2025
[4]

A millennium project: constructing small groups

H. U. Besche, B. Eick, and E. A. O’Brien. “A millennium project: constructing small groups”. In:International Jour- nal of Algebra and Computation12.05 (Oct. 2002), pp. 623–644.issn: 1793-6500.doi:10.1142/s0218196702001115. url:http://dx.doi.org/10.1142/S0218196702001115

doi:10.1142/s0218196702001115 2002
[5]

The enumeration of finite rings

S. R. Blackburn and K. R. McLean. “The enumeration of finite rings”. In:Journal of the London Mathematical Society106.4 (July 2022), pp. 3240–3262.issn: 1469-7750.doi:10.1112/jlms.12661.url:http://dx.doi.org/ 10.1112/jlms.12661

work page doi:10.1112/jlms.12661.url:http://dx.doi.org/ 2022
[6]

Cube-Based Isomorph-Free Finite Model Finding

C. Chow, M. Janota, and J. Ara´ ujo. “Cube-Based Isomorph-Free Finite Model Finding”. In:ECAI 2024. IOS Press, Oct. 2024.isbn: 9781643685489.doi:10.3233/faia240980.url:http://dx.doi.org/10.3233/FAIA240980

work page doi:10.3233/faia240980.url:http://dx.doi.org/10.3233/faia240980 2024
[7]

Contributors.MathStructures

A. Contributors.MathStructures. [Online; accessed 3-July-2025]. 2025.url:https://math.chapman.edu/ ~jipsen/ structures/doku.php?id=start&rev=1728801324

work page 2025
[8]

Contributors.semirings — MathStructures

A. Contributors.semirings — MathStructures. [Online; accessed 4-July-2025]. 2021.url:https://math.chapman. edu/~jipsen/structures/doku.php?id=semirings&rev=1614028296. 4 n up to isomorphism no additional constraints with 0 with 1 with 0 + 1 1 1 1 1 1 2 4 2 2 1 3 29 8 9 3 4 289 57 55 16 5 3,589 550 437 100 6 53,661 6,639 4,296 794 7 949,843 96,264 52,043 7...

work page 2025

Show all 48 references

[9]

Contributors.semirings with identity — MathStructures

A. Contributors.semirings with identity — MathStructures. [Online; accessed 4-July-2025]. 2021.url:https:// math.chapman.edu/~jipsen/structures/doku.php?id=semirings_with_identity&rev=1614028296

2025
[10]

Contributors.semirings with identity and zero — MathStructures

A. Contributors.semirings with identity and zero — MathStructures. [Online; accessed 4-July-2025]. 2021.url: https://math.chapman.edu/ ~jipsen/structures/doku.php?id=semirings_with_identity_and_zero&rev= 1614028296

2025
[11]

Distler and J

A. Distler and J. Mitchell.Smallsemi, A library of small semigroups, Version 0.7.2. GAP package. Feb. 2025.url: https://gap-packages.github.io/smallsemi/

2025
[12]

Classification and enumeration of finite semigroups

A. Distler. “Classification and enumeration of finite semigroups”. CC BY-NC-SA 3.0. PhD thesis. University of St Andrews, May 2010.url:https://hdl.handle.net/10023/945

2010
[13]

The semigroups of order 9 and their automorphism groups

A. Distler and T. Kelsey. “The semigroups of order 9 and their automorphism groups”. In:Semigroup Forum88.1 (June 2013), pp. 93–112.issn: 1432-2137.doi:10.1007/s00233-013-9504-9.url:http://dx.doi.org/10.1007/ s00233-013-9504-9

2013 doi
[14]

The Number of Nilpotent Semigroups of Degree 3

A. Distler and J. D. Mitchell. “The Number of Nilpotent Semigroups of Degree 3”. In:The Electronic Journal of Combinatorics19.2 (June 2012).issn: 1077-8926.doi:10.37236/2441.url:http://dx.doi.org/10.37236/2441

2012 doi
[15]

The Semigroups of Order 10

A. Distler et al. “The Semigroups of Order 10”. In:Principles and Practice of Constraint Programming. Springer Berlin Heidelberg, 2012, pp. 883–899.isbn: 9783642335587.doi:10.1007/978- 3- 642- 33558- 7_63.url:http: //dx.doi.org/10.1007/978-3-642-33558-7_63

2012 doi
[16]

Edwards, J

J. Edwards, J. D. Mitchell, and P. Ragavan.Semirings, Counting and enumerating semirings Version 0.2.0. GAP package. July 2025.url:https://github.com/pramothragavan/semirings

2025
[17]

Classification of Finite Rings of Orderp 2

B. Fine. “Classification of Finite Rings of Orderp 2”. In:Mathematics Magazine66.4 (Oct. 1993), p. 248.issn: 0025-570X.doi:10.2307/2690742.url:http://dx.doi.org/10.2307/2690742

1993 doi
[18]

SW AC Computes 126 Distinct Semigroups of Order 4

G. E. Forsythe. “SW AC Computes 126 Distinct Semigroups of Order 4”. In:Proceedings of the American Mathe- matical Society6.3 (June 1955), p. 443.issn: 0002-9939.doi:10.2307/2032786.url:http://dx.doi.org/10. 2307/2032786. [19]GAP – Groups, Algorithms, and Programming, Version ...

1955 doi
[20]

Computing finite commutative semigroups

P. A. Grillet. “Computing finite commutative semigroups”. In:Semigroup Forum53.1 (Dec. 1996), pp. 140–154. issn: 1432-2137.doi:10.1007/bf02574129.url:http://dx.doi.org/10.1007/BF02574129

1996 doi
[21]

Counting Semigroups

P. A. Grillet. “Counting Semigroups”. In:Communications in Algebra43.2 (Aug. 2014), pp. 574–596.issn: 1532- 4125.doi:10.1080/00927872.2013.790036.url:http://dx.doi.org/10.1080/00927872.2013.790036

2014
[22]

D. F. Holt, B. Eick, and E. A. O’Brien.Handbook of Computational Group Theory. Chapman and Hall/CRC, Jan. 2005.isbn: 9780429147944.doi:10.1201/9781420035216.url:http://dx.doi.org/10.1201/9781420035216

2005 doi
[23]

Junttila and P

T. Junttila and P. Kaski.bliss: A Tool for Computing Automorphism Groups and Canonical Labelings of Graphs. Version as of July 2025. 2007.url:https://users.aalto.fi/ ~tjunttil/bliss/

2025
[24]

Engineering an efficient canonical labeling tool for large and sparse graphs

T. Junttila and P. Kaski. “Engineering an efficient canonical labeling tool for large and sparse graphs”. In:Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments (ALENEX)(2007), pp. 135–149.doi:10.1137/ 1.9781611972870.13.url:https://epubs.siam.org/doi/10...

2007 doi
[25]

Die halbgruppen der ordnungen≤7

H. J¨ urgensen, P. Wick, and V. von ˇSt. Schwarz. “Die halbgruppen der ordnungen≤7”. In:Semigroup Forum 14.1 (Dec. 1977), pp. 69–79.issn: 1432-2137.doi:10.1007/bf02194655.url:http://dx.doi.org/10.1007/ BF02194655

1977 doi
[26]

The November meeting in Los Angeles

V. L. Klee Jr. “The November meeting in Los Angeles”. In:Bull. Amer. Math. Soc.64.2 (1958), p. 56.issn: 0002- 9904.doi:10.1090/S0002-9904-1958-10159-7.url:https://doi.org/10.1090/S0002-9904-1958-10159-7

1958 doi
[27]

The Number of Semigroups of Ordern

D. J. Kleitman, B. R. Rothschild, and J. H. Spencer. “The Number of Semigroups of Ordern”. In:Proceedings of the American Mathematical Society55.1 (Feb. 1976), p. 227.issn: 0002-9939.doi:10 . 2307 / 2041879.url: http://dx.doi.org/10.2307/2041879

1976 doi
[28]

Enumerating Finite Rings

R. L. Kruse and D. T. Price. “Enumerating Finite Rings”. In:Journal of the London Mathematical Societys2-2.1 (Jan. 1970), pp. 149–159.issn: 0024-6107.doi:10.1112/jlms/s2-2.1.149.url:http://dx.doi.org/10.1112/ jlms/s2-2.1.149

1970 doi
[29]

Lothaire.Applied Combinatorics on Words

M. Lothaire.Applied Combinatorics on Words. Cambridge University Press, July 2005.isbn: 9781107341005.doi: 10.1017/cbo9781107341005.url:http://dx.doi.org/10.1017/CBO9781107341005

2005 doi
[30]

Enumeration of finite inverse semigroups

M. E. Malandro. “Enumeration of finite inverse semigroups”. In:Semigroup Forum99.3 (Sept. 2019), pp. 679–723. issn: 1432-2137.doi:10.1007/s00233-019-10054-9.url:http://dx.doi.org/10.1007/s00233-019-10054-9

2019 doi
[31]

Graph isomorphism, general remarks

G. L. Miller. “Graph isomorphism, general remarks”. In:Proceedings of the ninth annual ACM symposium on Theory of computing - STOC ’77. STOC ’77. ACM Press, 1977, pp. 143–150.doi:10.1145/800105.803404.url: http://dx.doi.org/10.1145/800105.803404

1977
[32]

J. D. Mitchell et al.Semigroups - GAP package, Version 5.5.1. June 2025.doi:10.5281/zenodo.592893.url: http://dx.doi.org/10.5281/zenodo.592893

2025 doi
[33]

Semigroups of Order Five

T. S. Motzkin and J. L. Selfridge. “Semigroups of Order Five”. In: Presented in [26]
[34]

Entry A234845 in The On-Line Ency- clopedia of Integer Sequences

OEIS Foundation Inc.Number of commutative inverse monoids of ordern. Entry A234845 in The On-Line Ency- clopedia of Integer Sequences. 2025.url:https://oeis.org/A234845

2025
[35]

Entry A234843 in The On-Line Encyclopedia of Integer Sequences

OEIS Foundation Inc.Number of commutative inverse semigroups of ordern. Entry A234843 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A234843

2025
[36]

Entry A234844 in The On-Line Encyclopedia of Integer Sequences

OEIS Foundation Inc.Number of inverse monoids of ordern. Entry A234844 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A234844

2025
[37]

2025.url:https://oeis.org/A001428

OEIS Foundation Inc.Number of inverse semigroups of ordern, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator), Entry A001428 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A001428

2025
[38]

2025.url:https://oeis.org/A058129

OEIS Foundation Inc.Number of nonisomorphic monoids (semigroups with identity) of ordern, Entry A058129 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A058129

2025
[39]

2025.url:https://oeis.org/A027851

OEIS Foundation Inc.Number of nonisomorphic semigroups of ordern, Entry A027851 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A027851

2025
[40]

2025.url:https://oeis.org/A001423

OEIS Foundation Inc.Number of semigroups of ordern, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator), Entry A001423 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A001423

2025
[41]

2025.url:https://oeis.org/A001426

OEIS Foundation Inc.The number of commutative semigroups of ordern, Entry A001426 in The On-Line Ency- clopedia of Integer Sequences. 2025.url:https://oeis.org/A001426

2025
[42]

2025.url:https://oeis.org/A000001

OEIS Foundation Inc.The number of groups of ordern, Entry A000001 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A000001

2025
[43]

2025.url:https://oeis.org/A305858

OEIS Foundation Inc.The number of near-rings withnelements, Entry A305858 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A305858

2025
[44]

2025.url:https://oeis.org/A027623

OEIS Foundation Inc.The number of rings withnelements, Entry A027623 in The On-Line Encyclopedia of Integer Sequences. 2025.url:https://oeis.org/A027623

2025
[45]

There are 15973 semigroups of order 6

R. J. Plemmons. “There are 15973 semigroups of order 6”. In:Math. Algorithms2 (1967), pp. 2–17.issn: 0025-5548

1967
[46]

The finite basis problem for additively idempotent semirings of order four, I

M. Ren et al. “The finite basis problem for additively idempotent semirings of order four, I”. In:Semigroup Forum 110.2 (Apr. 2025), pp. 422–457.issn: 1432-2137.doi:10.1007/s00233-025-10520-7.url:http://dx.doi.org/ 10.1007/s00233-025-10520-7. 6

2025 doi
[47]

Sakarovitch.Elements of Automata Theory

J. Sakarovitch.Elements of Automata Theory. Ed. by R. Thomas. Cambridge University Press, Oct. 2009.isbn: 9781139195218.doi:10.1017/cbo9781139195218.url:http://dx.doi.org/10.1017/CBO9781139195218

2009 doi
[48]

Semigroups of order 8

S. Satoh, K. Yama, and M. Tokizawa. “Semigroups of order 8”. In:Semigroup Forum49.1 (Dec. 1994), pp. 7–29. issn: 1432-2137.doi:10.1007/bf02573467.url:http://dx.doi.org/10.1007/BF02573467. [49]Semirings of small orders.https : / / math . stackexchange . com / questions / 396101...

1994 doi
[50]

The variety generated by an ai-semiring of order three

X. Zhao et al. “The variety generated by an ai-semiring of order three”. In:Ural Mathematical Journal6.2 (Dec. 2020), p. 117.issn: 2414-3952.doi:10.15826/umj.2020.2.012.url:http://dx.doi.org/10.15826/umj.2020. 2.012. 7

2020 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Aichinger et al.SONATA, System of nearrings and their applications, Version 2.9.6

E. Aichinger et al.SONATA, System of nearrings and their applications, Version 2.9.6. Refereed GAP package. Dec. 2022.url:https://gap-packages.github.io/sonata/

work page 2022

[2] [2]

Bauer.alg: Algebraic library for constructive mathematics.https://github.com/andrejbauer/alg

A. Bauer.alg: Algebraic library for constructive mathematics.https://github.com/andrejbauer/alg. Accessed: 2025-07-03. 2019

work page 2025

[3] [3]

Bauer.Issue #16: Incorrect results.https://github.com/andrejbauer/alg/issues/16

A. Bauer.Issue #16: Incorrect results.https://github.com/andrejbauer/alg/issues/16. GitHub issue, opened Jul 4, 2025. 2025

work page 2025

[4] [4]

A millennium project: constructing small groups

H. U. Besche, B. Eick, and E. A. O’Brien. “A millennium project: constructing small groups”. In:International Jour- nal of Algebra and Computation12.05 (Oct. 2002), pp. 623–644.issn: 1793-6500.doi:10.1142/s0218196702001115. url:http://dx.doi.org/10.1142/S0218196702001115

doi:10.1142/s0218196702001115 2002

[5] [5]

The enumeration of finite rings

S. R. Blackburn and K. R. McLean. “The enumeration of finite rings”. In:Journal of the London Mathematical Society106.4 (July 2022), pp. 3240–3262.issn: 1469-7750.doi:10.1112/jlms.12661.url:http://dx.doi.org/ 10.1112/jlms.12661

work page doi:10.1112/jlms.12661.url:http://dx.doi.org/ 2022

[6] [6]

Cube-Based Isomorph-Free Finite Model Finding

C. Chow, M. Janota, and J. Ara´ ujo. “Cube-Based Isomorph-Free Finite Model Finding”. In:ECAI 2024. IOS Press, Oct. 2024.isbn: 9781643685489.doi:10.3233/faia240980.url:http://dx.doi.org/10.3233/FAIA240980

work page doi:10.3233/faia240980.url:http://dx.doi.org/10.3233/faia240980 2024

[7] [7]

Contributors.MathStructures

A. Contributors.MathStructures. [Online; accessed 3-July-2025]. 2025.url:https://math.chapman.edu/ ~jipsen/ structures/doku.php?id=start&rev=1728801324

work page 2025

[8] [8]

Contributors.semirings — MathStructures

A. Contributors.semirings — MathStructures. [Online; accessed 4-July-2025]. 2021.url:https://math.chapman. edu/~jipsen/structures/doku.php?id=semirings&rev=1614028296. 4 n up to isomorphism no additional constraints with 0 with 1 with 0 + 1 1 1 1 1 1 2 4 2 2 1 3 29 8 9 3 4 289 57 55 16 5 3,589 550 437 100 6 53,661 6,639 4,296 794 7 949,843 96,264 52,043 7...

work page 2025