Pith. sign in

REVIEW 1 major objections 24 references

Operad of posets 101: The Wix\'arika posets

T0 review · 1 major / 0 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Wixárica posets form a suboperad of the operad of all posets together with their algebras.

desk verdict Wixárica posets supply a named concrete suboperad example, but the closure under composition is asserted rather than shown in the abstract. read the letter →

arxiv 2406.07370 v3 submitted 2024-06-11 math.CO

José Antonio Arciniega-Nevárez , Marko Berghoff , Eric Rubiel Dolores-Cuenca This is my paper
classification math.CO
keywords operadsposetssuboperadsWixáricacombinatoricsoperadalgebrasposetcompositionssuboperadexamples
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 defines a concrete family of posets called Wixárica posets and verifies that this family stays closed under the composition rules that make all posets into an operad. The closure turns the family into its own smaller operad, so one can form algebras over it that encode combinatorial data respecting those rules. A reader cares because the construction supplies an explicit, computable setting in which to see how poset combinatorics organizes into algebraic structures without first mastering the most general theory. The note selects this family precisely because it is rich enough to display the main structural features yet simple enough to check by hand.

What carries the argument

The Wixárica posets, a specific family of posets that is closed under the operad composition maps and contains the unit, thereby forming a suboperad.

What would settle it

Direct computation of the operad composition of two Wixárica posets that yields a poset outside the Wixárica family would falsify the suboperad claim.

Watch

Extended reading notes

Core claim

We present a nontrivial example of a suboperad of the operad of posets, called Wixárica posets, together with its associated algebras. This example is sufficiently rich to exhibit key structural features of the theory, while remaining accessible and avoiding unnecessary technicalities.

Load-bearing premise

The defined Wixárica family must remain inside itself when any two members are composed using the operad composition of posets.

Editorial extensions

If this is right

  • Algebras over the Wixárica suboperad give combinatorial models for structures that respect the restricted compositions.
  • Iterated compositions of Wixárica posets stay inside the family, supporting recursive constructions.
  • Structural properties of operads, such as associativity of composition, can be checked explicitly on this example.
  • The associated algebras provide a setting in which to study morphisms and representations tied directly to the poset family.

Reading between the lines

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

  • Other naturally occurring families of posets may likewise close under the same composition rules and yield further suboperads.
  • The algebras could be used to count or classify ordered objects that admit a restricted form of hierarchical assembly.
  • Explicit generators or relations for the Wixárica suboperad might be derivable from the poset definitions, opening a route to a presentation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript presents Wixárica posets as a nontrivial suboperad of the operad of posets, together with the associated algebras. It frames this as an accessible example that exhibits key structural features of operad theory applied to poset combinatorics while avoiding unnecessary technicalities.

Significance. If the suboperad property is established, the construction supplies a concrete, low-technicality example of a suboperad and its algebras inside the poset operad; such examples are useful for illustrating composition and algebra structures in combinatorial operad theory.

major comments (1)
  1. [Abstract] Abstract (and any subsequent definition section): the central claim that the Wixárica family forms a suboperad requires explicit verification that the family is closed under the operad composition maps of the ambient poset operad and contains the unit. The manuscript must define the family and either exhibit the closure check or supply an inductive argument on the defining properties; without this step the suboperad assertion remains unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the need for explicit verification of the suboperad axioms. We agree that the manuscript should contain a self-contained check of closure and unit preservation rather than relying on the reader to reconstruct it from the definitions. The revision will add this material without altering the overall accessibility of the note.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and any subsequent definition section): the central claim that the Wixárica family forms a suboperad requires explicit verification that the family is closed under the operad composition maps of the ambient poset operad and contains the unit. The manuscript must define the family and either exhibit the closure check or supply an inductive argument on the defining properties; without this step the suboperad assertion remains unverified.

    Authors: We accept the point. The current draft defines the Wixárica family via a recursive combinatorial condition on labeled posets and states that it is closed under the poset operad composition, but does not spell out the verification. In the revised manuscript we will insert, immediately after the definition, a short subsection that (i) recalls the unit (the singleton poset) and verifies it satisfies the Wixárica condition, and (ii) proves closure by a direct case analysis on how the composition identifies the maximal element of the first poset with the minimal element of the second, showing that the resulting labeled poset again obeys the Wixárica recursion. The argument is elementary and does not require additional technical machinery. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Wixárica posets introduced as independently defined suboperad example

full rationale

The paper presents a new combinatorial family (Wixárica posets) claimed to form a suboperad inside the operad of all posets. No equations, definitions, or citations in the provided abstract reduce the suboperad property to a fitted parameter, self-citation chain, or definitional tautology. The central step—closure under operadic composition—is described as a direct verification on the explicitly defined objects rather than an input that is renamed as output. This matches the default case of a self-contained example construction with no load-bearing reduction to its own inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The claim rests on the standard axioms of operads (associativity of composition, unit laws) and on the combinatorial definition of the new Wixárica family; no free parameters or invented physical entities appear.

assumptions (1)
  • standard math Operad axioms (associativity, units) hold for the ambient operad of posets
    Invoked when asserting that the Wixárica family is a suboperad.
invented entities (1)
  • Wixárica posets
    purpose: Concrete family of posets closed under operad composition
    Newly defined class whose closure properties constitute the central example.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Operad of posets 101: The Wix\'arika posets." pith.science (2026). https://pith.science/paper/2406.07370

@misc{pith2026240607370,
  author       = {Pith},
  title        = {Pith review of: Operad of posets 101: The Wix\'arika posets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2406.07370}},
  note         = {Machine review of arXiv:2406.07370}
}
read the original abstract

We study classes of objects whose combinatorics are closely related to those of posets. The framework of operads and operad algebras allows us to make this relationship precise and provides tools for a deeper understanding of their combinatorial structure. In this note, we present a nontrivial example of a suboperad of the operad of posets, called Wix\'arika posets, together with its associated algebras. This example is sufficiently rich to exhibit key structural features of the theory, while remaining accessible and avoiding unnecessary technicalities.

Figures

Figures reproduced from arXiv: 2406.07370 by the authors.

Figure 1
Figure 1. To the left bottom we have a Wixarika poset, and above it a decomposition in terms of ∗, D. To the right we have the tree of operations. The first step of our algorithm, replacement of the leaves ⟨1⟩ by their order series Z (1) = x (1−x) 2 , followed by evaluation of the operations, returns the power series D(D(D(D(1) ∗ Z (1) ∗ D(1))) ∗ Z (3)) = D(D(D((Z (3) + 2Z (4)) ∗ Z (1) ∗ (Z (3) + 2Z (4)))) ∗ Z (3)) = D(D(D(Z … view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Shuffle series, 2023

    Khushdil Ahmad, Eric Rubiel Dolores-Cuenca, and Khurram Shabbir. Shuffle series, 2023

    work page 2023

  2. [2]

    An algebra over the operad of posets and structural binomial identities

    Jos \'e Antonio Arciniega-Nev \'a rez, Marko Berghoff, and Eric Rubiel Dolores-Cuenca. An algebra over the operad of posets and structural binomial identities. Bol. Soc. Mat. Mex., III. Ser. , 29(1):29, 2023. Id/No 8

    work page 2023

  3. [3]

    Bruce C. Berndt. Ramanujan’s Notebooks, Part I . Springer, New York, NY, 1 edition, 1985

    work page 1985

  4. [4]

    Doppelgangers: the U r-operation and posets of bounded height, 2018

    Thomas Browning, Max Hopkins, and Zander Kelley. Doppelgangers: the U r-operation and posets of bounded height, 2018

    work page 2018

  5. [5]

    Combinatorial Species and Tree-like Structures

    François Bergeron, Gilbert Labelle, and Pierre Leroux. Combinatorial Species and Tree-like Structures . Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1997

    work page 1997

  6. [6]

    An Ehrhart series formula for reflexive polytopes

    Benjamin Braun. An Ehrhart series formula for reflexive polytopes. Electron. J. Comb. , 13, 2006

    work page 2006

  7. [7]

    Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics , volume 195

    Matthias Beck and Raman Sanyal. Combinatorial Reciprocity Theorems: An Invitation to Enumerative Geometric Combinatorics , volume 195. American Mathematical Society, Providence, Rhode Island, first edition, 2018

    work page 2018

  8. [8]

    Operads and algebraic combinatorics of trees

    Fr \'e d \'e ric Chapoton. Operads and algebraic combinatorics of trees. S \'e min. Lothar. Comb. , 58:b58c, 27, 2007

    work page 2007

Show all 24 references
  1. [9]

    Mendoza-Cortes

    Eric Rubiel Dolores-cuenca and Jose L. Mendoza-Cortes. A poset version of ramanujan results on eulerian numbers and zeta values, 2024

    2024

  2. [10]

    Sur les poly \`e dres rationnels homoth \'e tiques \`a \(n\) dimensions

    Eug \`e ne Ehrhart. Sur les poly \`e dres rationnels homoth \'e tiques \`a \(n\) dimensions. C. R. Acad. Sci., Paris , 254:616--618, 1962

    1962

  3. [11]

    L. Euler. Institutiones Calculi differentialis . Acad. Imperialis Sci., Petrograd, 1755

  4. [12]

    Faigle and R

    U. Faigle and R. Schrader. On the computational complexity of the order polynomial. Discrete Appl. Math. , 15:261--269, 1986

    1986

  5. [13]

    Classification of ehrhart polynomials of integral simplices

    Akihiro Higashitani. Classification of ehrhart polynomials of integral simplices

  6. [14]

    Simplicial and dendroidal homotopy theory , volume 75 of Ergeb

    Gijs Heuts and Ieke Moerdijk. Simplicial and dendroidal homotopy theory , volume 75 of Ergeb. Math. Grenzgeb., 3. Folge . Cham: Springer, 2022

    2022

  7. [15]

    Doppelgängers: Bijections of Plane Partitions

    Zachary Hamaker, Rebecca Patrias, Oliver Pechenik, and Nathan Williams. Doppelgängers: Bijections of Plane Partitions . International Mathematics Research Notices , 2020(2):487--540, 03 2018

    2020

  8. [16]

    Algebraic Operads

    Jean-Louis Loday and Bruno Vallette. Algebraic Operads . Springer-Verlag Berlin Heidelberg, 1st edition, 2012

    2012

  9. [17]

    What are operads? Resonance , 25:397--417, 2020

    Anita Naolekar. What are operads? Resonance , 25:397--417, 2020

    2020

  10. [18]

    okovi \'c

    Dragomir Z . okovi \'c . Summation of certain types of series. Publications de l'Institut Math\'ematique , 4(18)(24):43--55, 1964

    1964

  11. [19]

    o der. Ordered sets. An introduction with connections from combinatorics to topology . Basel: Birkh \

    Bernd Schr \"o der. Ordered sets. An introduction with connections from combinatorics to topology . Basel: Birkh \"a user/Springer, 2nd edition edition, 2016

    2016

  12. [20]

    R. P. Stanley. A chromatic-like polynomial for ordered sets. In Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications , pages 421--427, May 1970

    1970

  13. [21]

    What is an operad? Notices Am

    Jim Stasheff. What is an operad? Notices Am. Math. Soc. , 51(6):630--631, 2004

    2004

  14. [22]

    William T. Trotter. Combinatorics and partially ordered sets: dimension theory . Johns Hopkins Ser. Math. Sci. Baltimore: The Johns Hopkins University Press, 1992

    1992

  15. [23]

    D. G. Wagner. Enumeration of functions from posets to chains. European Journal of Combinatorics , 13(4):313--324, 1992

    1992

  16. [24]

    \(D\) -log and formal flow for analytic isomorphisms of \(n\) -space

    David Wright and Wenhua Zhao. \(D\) -log and formal flow for analytic isomorphisms of \(n\) -space. Trans. Am. Math. Soc. , 355(8):3117--3141, 2003

    2003

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.