pith. sign in

arxiv: 2602.15754 · v3 · pith:M4OJVJBPnew · submitted 2026-02-17 · 🧮 math.RA · math.CO· math.NT

Power monoids and their arithmetic: a survey

Salvatore Tringali This is my paper

Pith reviewed 2026-05-21 12:15 UTC · model grok-4.3

classification 🧮 math.RA math.COmath.NT
keywords power monoidsfactorization theorymonoidssetwise multiplicationnon-cancellativenon-commutativearithmetic properties
0
0 comments X

The pith

Non-empty finite subsets of a monoid form a monoid under setwise multiplication, with unusual arithmetic properties useful for factorization studies in non-cancellative or non-commutative settings.

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

This survey examines power monoids, the monoids formed by non-empty finite subsets of a given multiplicatively written monoid under the operation of setwise multiplication, along with the variant that requires the subsets to contain the identity. The paper reviews recent developments showing that these structures display arithmetic behaviors distinct from those of the underlying monoid. A sympathetic reader would care because the work extends factorization theory to algebraic settings where cancellation or commutativity cannot be assumed.

Core claim

Power monoids are the monoids consisting of the non-empty finite subsets of a multiplicatively written monoid equipped with setwise multiplication; the same holds when restricting to subsets that contain the identity element. These constructions possess arithmetic properties that differ from the base monoid and thereby open new perspectives on the study of factorizations in non-cancellative or non-commutative monoids. The survey collects and organizes recent advances in this area.

What carries the argument

Power monoids formed by non-empty finite subsets under setwise multiplication

Load-bearing premise

The survey assumes that the selected recent developments represent the most relevant advances worth reviewing, without providing a systematic justification for literature inclusion criteria.

What would settle it

A concrete monoid in which the power monoid exhibits exactly the same factorization lengths and irreducibility patterns as the original monoid would challenge the emphasis on unusual arithmetic properties.

read the original abstract

The non-empty finite subsets of a multiplicatively written monoid form a monoid under setwise multiplication. The same holds for finite subsets containing the identity element. Partly due to their unusual arithmetic properties, these structures, generically known as power monoids, have attracted increasing attention in recent years, stimulating new perspectives in the study of factorizations in non-cancellative or non-commutative settings. We survey these developments and briefly review some related aspects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript is a survey on power monoids, defined as the monoids of non-empty finite subsets of a multiplicatively written monoid under setwise multiplication (and the variant consisting of finite subsets containing the identity). It recalls their arithmetic properties and surveys recent developments showing how these structures yield new perspectives on factorization theory in non-cancellative or non-commutative settings.

Significance. If the survey is representative, it would consolidate an emerging area of monoid arithmetic and help researchers access recent advances in factorization outside classical settings. The paper performs the useful service of collecting and organizing results that are currently scattered.

major comments (1)
  1. [Introduction] Introduction: the central claim that power monoids 'have attracted increasing attention in recent years, stimulating new perspectives' is not accompanied by any stated inclusion criteria, time window, search protocol, or completeness argument for the literature reviewed. This is load-bearing for a survey whose value rests on representing the relevant developments.
minor comments (1)
  1. [Abstract] The abstract states that the paper will 'briefly review some related aspects' without indicating what those aspects are; a short enumeration would improve reader orientation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting an important point about the presentation of our survey. We address the major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Introduction] Introduction: the central claim that power monoids 'have attracted increasing attention in recent years, stimulating new perspectives' is not accompanied by any stated inclusion criteria, time window, search protocol, or completeness argument for the literature reviewed. This is load-bearing for a survey whose value rests on representing the relevant developments.

    Authors: We agree that explicitly documenting the scope of the literature review would improve transparency and credibility, particularly for a survey on an emerging topic. In the revised version we will add a short subsection (or expanded paragraph) at the end of the Introduction that states: our review focuses on developments from approximately 2015 onward; the cited works were identified via searches on arXiv, MathSciNet and Google Scholar using the keywords 'power monoid', 'power set monoid' and 'factorization theory non-cancellative monoids'; the field remains small and concentrated, so the survey aims to cover the principal lines of research rather than claim exhaustive coverage. This addition will directly support the claim of increasing attention by referencing the growth in publications while clarifying the selection process. revision: yes

Circularity Check

0 steps flagged

No circularity in survey structure or claims

full rationale

This is a survey paper that reviews existing developments in power monoids without presenting original derivations, predictions, fitted parameters, or first-principles results. The abstract and described content rely on external citations to prior work rather than any internal chain that reduces to its own inputs by construction. No equations, uniqueness theorems, or ansatzes are invoked in a self-referential manner, and the selection of literature is framed as a review without claiming to derive new results from the surveyed material itself. The paper is therefore self-contained against external benchmarks with no load-bearing steps that exhibit circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey paper the work does not introduce new free parameters, axioms, or invented entities; it reviews established concepts from monoid theory.

pith-pipeline@v0.9.0 · 5587 in / 927 out tokens · 34958 ms · 2026-05-21T12:15:05.722255+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the automorphisms of the power semigroups of a numerical semigroup

    math.NT 2026-04 unverdicted novelty 6.0

    The automorphism group of the power semigroup P(H) of any numerical semigroup H is trivial.

Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Aggarwal, F

    A. Aggarwal, F. Gotti, and S. Lu,On finitary power monoids of linearly orderable monoids, preprint (arXiv:2412.05857)

    work page arXiv

  2. [2]

    Ajran and F

    K. Ajran and F. Gotti,Factoriality inside Boolean lattices, preprint (arXiv:2305.00413)

    work page arXiv

  3. [3]

    Almeida,Some Key Problems on Finite Semigroups, Semigroup Forum64(2002), 159–179

    J. Almeida,Some Key Problems on Finite Semigroups, Semigroup Forum64(2002), 159–179

    work page 2002

  4. [4]

    Bounded and finite factorization domains

    D. F. Anderson and F. Gotti, “Bounded and finite factorization domains”, pp. 7–57 in: A. Badawi and J. Coykendall,Rings, Monoids, and Module Theory, Springer Proc. Math. Stat.382, Springer, 2022

    work page 2022

  5. [5]

    A. A. Antoniou and S. Tringali,On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups, Pacific J. Math.312(2021), No. 2, 279–308

    work page 2021

  6. [6]

    Baginski and S

    P. Baginski and S. T. Chapman,Factorizations of algebraic integers, block monoids, and additive number theory, Amer. Math. Monthly118(2011), No. 10, 901–920

    work page 2011

  7. [7]

    Bienvenu and A

    P.-Y . Bienvenu and A. Geroldinger,On algebraic properties of power monoids of numerical monoids, Israel J. Math.265(2025), 867–900

    work page 2025

  8. [8]

    Bonzio and P

    S. Bonzio and P. García-Sánchez,When the poset of the ideal class monoid of a numerical semigroup is a lattice, preprint (arXiv:2412.07281)

    work page arXiv

  9. [9]

    Casabella, M

    L. Casabella, M. D’Anna, and P. García–Sánchez,Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup, Mediterr. J. Math.21(2024), Art. 7

    work page 2024

  10. [10]

    Some Applications of a New Approach to Factorization

    L. Cossu, “Some Applications of a New Approach to Factorization”, pp. 73–94 in: M. Brešar, A. Geroldinger, B. Olberding, and D. Smertnig (eds.),Recent Progress in Ring and Factorization Theory (Graz, Austria, July 10–14, 2023), Springer Proc. Math. Stat.477, Springer, 2025

    work page 2023

  11. [11]

    Cossu and S

    L. Cossu and S. Tringali,Factorization under Local Finiteness Conditions, J. Algebra630(2023), 128– 161

    work page 2023

  12. [12]

    Cossu and S

    L. Cossu and S. Tringali,Abstract Factorization Theorems with Applications to Idempotent Factoriza- tions, Israel J. Math.263(2024), 349–395

    work page 2024

  13. [13]

    Cossu and S

    L. Cossu and S. Tringali,On the finiteness of certain factorization invariants, Ark. för Mat.62(2024), No. 1, 21–38

    work page 2024

  14. [14]

    Cossu and S

    L. Cossu and S. Tringali,On the arithmetic of power monoids, J. Algebra686(Jan 2026), 793–813

    work page 2026

  15. [15]

    J. Dani, F. Gotti, L. Hong, B. Li, and S. Schlessinger,On finitary power monoids of linearly orderable monoids, preprint (arXiv:2501.03407)

    work page arXiv

  16. [16]

    Dubreil,Contribution à la théorie des demi-groupes

    P. Dubreil,Contribution à la théorie des demi-groupes. III, Bull. Soc. Math. France81(1953), 289–306

    work page 1953

  17. [17]

    Fan and S

    Y . Fan and S. Tringali,Power monoids: A bridge between Factorization Theory and Arithmetic Combi- natorics, J. Algebra512(Oct. 2018), 252–294

    work page 2018

  18. [18]

    Gan and X

    A. Gan and X. Zhao,Global determinism of Clifford semigroups, J. Australian Math. Soc.97(2014), 63–77

    work page 2014

  19. [19]

    P. A. García-Sánchez and C. O’Neill,Lengths of irreducible decompositions of numerical semigroups, preprint (arXiv:2602.00404)

    work page arXiv

  20. [20]

    P. A. García-Sánchez and S. Tringali,Semigroups of ideals and isomorphism problems, Proc. Amer. Math. Soc.153(2025), No. 6, 2323–2339

    work page 2025

  21. [21]

    Geroldinger,Sets of lengths, Amer

    A. Geroldinger,Sets of lengths, Amer. Math. Monthly123(2016), 960–988

    work page 2016

  22. [22]

    Geroldinger and F

    A. Geroldinger and F. Halter-Koch,Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math.278, Chapman & Hall/CRC, 2006

    work page 2006

  23. [23]

    Geroldinger and Q

    A. Geroldinger and Q. Zhong,Factorization theory in commutative monoids, Semigroup Forum100 (2020), 22–51

    work page 2020

  24. [24]

    Gonzalez, E

    V . Gonzalez, E. Li, H. Rabinovitz, P. Rodriguez, and M. Tirador,On the atomicity of power monoids of Puiseux monoids, Internat. J. Algebra Comput.35(2025), No. 2, 167–181

    work page 2025

  25. [25]

    H. B. Hamilton and T. E. Nordahl,Tribute for Takayuki Tamura on his 90th birthday, Semigroup Forum 79(2009), 2–14

    work page 2009

  26. [26]

    J. C. Higgins,RepresentingN-semigroups, Bull. Austral. Math. Soc.1(1969), 115–125

    work page 1969

  27. [27]

    J. M. Howie,Fundamentals of Semigroup Theory, London Math. Soc. Monogr. Ser.12, Oxford Univ. Press, 1995

    work page 1995

  28. [28]

    J. H. B. Kemperman,On complexes in a semigroup, Indag. Math. (N.S.)18(1956), 247–254

    work page 1956

  29. [29]

    Kobayashi,Semilattices are globally determined, Semigroup Forum29(1984), No

    Y . Kobayashi,Semilattices are globally determined, Semigroup Forum29(1984), No. 1, 217–222

    work page 1984

  30. [30]

    E. M. Mogiljanskaja,Non-isomorphic semigroups with isomorphic semigroups of subsets, Semigroup Forum6(1973), 330–333

    work page 1973

  31. [31]

    Ostmann,Additive Zahlentheorie, 1

    H.-H. Ostmann,Additive Zahlentheorie, 1. Teil: Allgemeine Untersuchungen, Springer-Verlag, 1968

    work page 1968

  32. [32]

    Rago,The isomorphism problem for reduced finitary power monoids, preprint (arXiv:2601.22469)

    B. Rago,The isomorphism problem for reduced finitary power monoids, preprint (arXiv:2601.22469)

    work page arXiv

  33. [33]

    Rago,The automorphism group of reduced power monoids of finite abelian groups, preprint (arXiv:2510.17533)

    B. Rago,The automorphism group of reduced power monoids of finite abelian groups, preprint (arXiv:2510.17533)

    work page arXiv

  34. [34]

    Rago,A counterexample to an isomorphism problem for power monoids, Proc

    B. Rago,A counterexample to an isomorphism problem for power monoids, Proc. Amer. Math. Soc., to appear (arXiv:2509.23818). ———————- 17

    work page arXiv

  35. [35]

    Rédei,Algebra, vol

    L. Rédei,Algebra, vol. 1, Int. Ser. Monogr. Pure Appl. Math.91, Pergamon Press, 1967

    work page 1967

  36. [36]

    Reinhart,On the system of length sets of power monoids, preprint (arXiv:2508.10209)

    A. Reinhart,On the system of length sets of power monoids, preprint (arXiv:2508.10209)

    work page arXiv

  37. [37]

    J. C. Rosales, P. A. García–Sánchez, and J. I. García–García,Atomic commutative monoids and their elasticity, Semigroup Forum 68(1) (2004) 64–86

    work page 2004

  38. [38]

    Sárközy,On additive decompositions of the set of quadratic residues modulop, Acta Arith.155(2012), No

    A. Sárközy,On additive decompositions of the set of quadratic residues modulop, Acta Arith.155(2012), No. 1, 41–51

    work page 2012

  39. [39]

    Shafer,Note on power semigroups, Math

    J. Shafer,Note on power semigroups, Math. Japon.12(1967), 32

    work page 1967

  40. [40]

    Smertnig,Divide and transfer: non-unique factorizations beyond commutativity, Amer

    D. Smertnig,Divide and transfer: non-unique factorizations beyond commutativity, Amer. Math. Monthly, to appear (arXiv:2602.06222)

    work page arXiv

  41. [41]

    On the recent results in the study of power semigroups

    T. Tamura, “On the recent results in the study of power semigroups”, pp. 191–200 in: S. M. Goberstein and P. M. Higgins (eds.),Semigroups and their applications, Reidel Publishing Company, 1987

    work page 1987

  42. [42]

    Tamura and J

    T. Tamura and J. Shafer,Power semigroups, Math. Japon.12(1967), 25–32;Errata, ibid.29(1984), No. 4, 679

    work page 1967

  43. [43]

    Tringali,Small doubling in ordered semigroups, Semigroup Forum90(2015), 135–148

    S. Tringali,Small doubling in ordered semigroups, Semigroup Forum90(2015), 135–148

    work page 2015

  44. [44]

    Tringali,An abstract factorization theorem and some applications, J

    S. Tringali,An abstract factorization theorem and some applications, J. Algebra602(July 2022), 352– 380

    work page 2022

  45. [45]

    Tringali,A characterisation of atomicity, Math

    S. Tringali,A characterisation of atomicity, Math. Proc. Cambridge Phil. Soc.175(2023), No. 2, 459– 465

    work page 2023

  46. [46]

    On the isomorphism problem for power semigroups

    S. Tringali, “On the isomorphism problem for power semigroups”, pp. 429–437 in: M. Brešar, A. Geroldinger, B. Olberding, and D. Smertnig (eds.),Recent Progress in Ring and Factorization Theory (Graz, Austria, July 10–14, 2023), Springer Proc. Math. Stat.477, Springer, 2025

    work page 2023

  47. [47]

    Tringali and K

    S. Tringali and K. Wen,The Additive Group of Integers and the Automorphisms of its Power Monoids, preprint (arXiv:2504.12566)

    work page arXiv

  48. [48]

    Tringali and W

    S. Tringali and W. Yan,A conjecture of Bienvenu and Geroldinger on power monoids, Proc. Amer. Math. Soc.153(2025), No. 3, 913–919

    work page 2025

  49. [49]

    Tringali and W

    S. Tringali and W. Yan,On power monoids and their automorphisms, J. Combin. Theory Ser. A209 (2025), 105961, 16 pp

    work page 2025

  50. [50]

    Tringali and W

    S. Tringali and W. Yan,Torsion groups and the Bienvenu–Geroldinger conjecture, Bull. London Math. Soc., to appear (arXiv:2601.19592)

    work page arXiv

  51. [51]

    D. Wong, S. Xu, C. Zhang, and J. Zhao,On automorphism groups of power semigroups over numerical semigroups or over numerical monoids, preprint (arXiv:2512.12606). SALV A TORE “SALVO” TRINGALIis a full professor in the School of Mathematical Sciences at Hebei Normal University (Shijiazhuang, China). His research lies at the intersection of arithmetic combi...

    work page internal anchor Pith review Pith/arXiv arXiv