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arxiv: 1907.06805 · v1 · pith:YDASC672new · submitted 2019-07-16 · 🧮 math.RA

Involutions on Incidence Algebras of Finite Posets

Ivan Gargate , Michael Gargate This is my paper

Pith reviewed 2026-05-24 20:58 UTC · model grok-4.3

classification 🧮 math.RA
keywords incidence algebrainvolutionsfinite posetsorder two elementscombinatorial enumerationposet structure
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The pith

Formulas count the involutions in incidence algebras of star, Y, and rhombus posets over finite fields of odd characteristic.

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

The paper supplies explicit formulas that give the number of elements of order 2 inside the incidence algebra of a finite poset when the coefficient field has odd characteristic. It derives these counts separately for three families of posets and then assembles the same ideas into an algorithm that works for an arbitrary finite poset. A reader would care because the incidence algebra converts the order relations of the poset into an associative algebra whose units and torsion elements reflect combinatorial symmetries of the underlying structure.

Core claim

We give various formulas to compute the number of all involutions in an incidence algebra I(X,K), where X is a finite poset of star, Y or rhombus shape and K is a finite field of characteristic different from 2. The same techniques yield an algorithm that calculates the number of involutions on any finite poset.

What carries the argument

Formulas that enumerate solutions to f squared equals the identity inside the incidence algebra, obtained by examining the values of f on the covering relations and intervals of the poset.

If this is right

  • For each listed family of posets the number of involutions is given by a closed expression involving only the cardinality of the field and the number of elements in the poset.
  • The same counting procedure extends without change to produce an algorithm that terminates for every finite poset.
  • Involution counts become computable without listing every basis element of the algebra once the poset shape is known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The algorithm makes it feasible to tabulate involution numbers across all posets of moderate size and look for patterns by size or height.
  • The same counting technique may apply to other torsion orders or to the unit group of the incidence algebra.

Load-bearing premise

The coefficient field must have characteristic different from 2.

What would settle it

Direct enumeration of all algebra elements for the smallest star poset over the field with three elements yields a count different from the formula.

read the original abstract

We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic different from 2. Using the techniques describing here we show an algorithm to calculate the number of involutions on any finite poset.

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

0 major / 3 minor

Summary. The paper derives explicit formulas for the number of involutions (elements f satisfying f ∗ f = δ) in the incidence algebra I(X, K) over a finite field K of characteristic ≠ 2, for X equal to the star poset, the Y poset, and rhombus posets; it further presents a recursive algorithm that computes the count for an arbitrary finite poset by decomposing according to order ideals generated by minimal elements and reducing to smaller instances.

Significance. If the derivations hold, the work supplies closed-form expressions obtained by direct enumeration of admissible values on comparable pairs and solution of the resulting quadratic equations, together with a recursion grounded in the definition of the convolution product. These constitute concrete, parameter-free counting results and a verifiable computational procedure for a class of algebras arising in poset combinatorics.

minor comments (3)
  1. The abstract and introduction refer to “Rhombuses” in plural; the manuscript should clarify whether this denotes a single rhombus shape or a family parameterized by size, and state the precise poset diagrams used in each case.
  2. Section 3 (or the algorithmic part) should include a small worked example on a poset with four or five elements to illustrate the recursion step and confirm that the base cases match the closed-form formulas given for the star and Y posets.
  3. Notation for the incidence algebra multiplication and the identity δ should be introduced once at the beginning and used consistently; currently the convolution symbol ∗ appears only after the first formula.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of our manuscript, including the accurate summary of our results on counting involutions in incidence algebras for the star, Y, and rhombus posets, as well as the recursive algorithm for general finite posets. The recommendation for minor revision is noted; however, no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation enumerates solutions to f ∗ f = δ directly from the definition of convolution in the incidence algebra, solving the resulting finite system of quadratic equations over K (char ≠ 2) for the star, Y, and rhombus posets; the general algorithm decomposes via order ideals into strictly smaller identical problems. No parameter is fitted and then renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. All steps remain self-contained algebraic counting with the single explicit hypothesis stated up front.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the sole explicit domain assumption is the field characteristic condition.

axioms (1)
  • domain assumption The incidence algebra is taken over a finite field of characteristic different from 2.
    Stated explicitly in the abstract as a prerequisite for the formulas and algorithm.

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Reference graph

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16 extracted references · 16 canonical work pages

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