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arxiv: 1907.02137 · v1 · pith:RLWK4DZHnew · submitted 2019-07-03 · 🧮 math.AC

Identit\'es pond\'er\'ees Peirce-\'evanescentes

Richard Varro This is my paper

Pith reviewed 2026-05-25 09:02 UTC · model grok-4.3

classification 🧮 math.AC
keywords Peirce-evanescent identitiesbaric algebrasmutation algebrasPeirce spectrumfree commutative nonassociative algebrarooted binary treesweighted identities
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The pith

Mutation algebras satisfy every Peirce-evanescent identity, so any subset of the base field K can appear as the Peirce spectrum of an algebra obeying one.

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

The paper introduces Peirce-evanescent baric identities, which are polynomial identities that baric algebras satisfy precisely when their associated Peirce polynomials are identically zero. It equips the free commutative nonassociative algebra generated by a set T with an algebraic system structure so that the classes of baric algebras obeying a fixed collection of identities behave like algebraic varieties. Rooted binary trees with labeled leaves are used to compute the Peirce polynomials. The central result is that every mutation algebra obeys all such identities. From this it follows that, for any field K and any subset S of K, there exists a K-algebra satisfying a Peirce-evanescent identity whose Peirce spectrum equals S. Explicit generators for homogeneous and non-homogeneous identities are constructed in both the univariate and multivariate cases.

Core claim

Mutation algebras satisfy all Peirce-evanescent identities. Therefore any subset of the field K can be realized as the Peirce spectrum of a K-algebra that obeys a Peirce-evanescent identity. The algebraic system structure on the free commutative nonassociative algebra generated by T supplies the necessary closure properties that make this conclusion possible.

What carries the argument

The algebraic system structure placed on the free commutative nonassociative algebra generated by a set T, which transfers variety-like properties to classes of baric algebras obeying given identities.

If this is right

  • Every mutation algebra obeys every Peirce-evanescent identity.
  • For any field K and any subset S of K there exists a K-algebra obeying a Peirce-evanescent identity whose Peirce spectrum is exactly S.
  • Generators for homogeneous and non-homogeneous Peirce-evanescent identities can be produced by explicit procedures in one or several variables.
  • Rooted binary trees with labeled leaves give a combinatorial representation of the Peirce polynomials associated with the identities.

Where Pith is reading between the lines

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

  • The same tree-based bookkeeping might be used to decide membership in the ideal of all Peirce-evanescent identities for a given variety.
  • The construction could be tested on small finite fields to produce algebras with prescribed Peirce spectra that are not mutation algebras.
  • The variety-like behavior might allow one to form free objects in the category of baric algebras satisfying a fixed Peirce-evanescent identity.

Load-bearing premise

The algebraic system structure on the free algebra actually endows the classes of baric algebras satisfying the identities with the required closure properties under the usual operations.

What would settle it

An explicit mutation algebra together with a concrete Peirce-evanescent identity that the algebra fails to satisfy.

read the original abstract

Peirce-evanescent baric identities are polynomial identities verified by baric algebras such that their Peirce polynomials are the null polynomial. In this paper procedures for constructing such homogeneous and non homogeneous identities are given. For this we define an algebraic system structure on the free commutative nonassociative algebra generated by a set T which provides for classes of baric algebras satisfying a given set of identities similar properties to those of the varieties of algebras. Rooted binary trees with labeled leaves are used to explain the Peirce polynomials. It is shown that the mutation algebras satisfy all Peirce-evanescent identities, it results from this that any part of the field K can be the Peirce spectrum of a K- algebra satisfying a Peirce-evanescent identity. We end by giving methods to obtain generators of homogeneous and non-homogeneous Peirce-evanescent identities that are applied in several univariate and multivariate cases.

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 / 2 minor

Summary. The manuscript defines Peirce-evanescent baric identities as polynomial identities satisfied by baric algebras whose associated Peirce polynomials are identically zero. It equips the free commutative nonassociative algebra generated by a set T with an algebraic-system structure that endows classes of baric algebras obeying a fixed set of such identities with closure properties analogous to those of varieties. Labeled rooted binary trees are used to compute Peirce polynomials explicitly. The central result is that every mutation algebra satisfies all Peirce-evanescent identities, from which it follows that an arbitrary subset of the base field K can arise as the Peirce spectrum of a K-algebra satisfying at least one such identity. Explicit generators for homogeneous and non-homogeneous identities are constructed and illustrated in several univariate and multivariate cases.

Significance. If the explicit constructions and tree-based computations are correct, the work supplies a systematic method for realizing prescribed Peirce spectra inside the class of baric algebras and transfers algebraic-system closure properties to these classes. The combinatorial representation via labeled trees and the direct verification on mutation algebras constitute concrete, checkable strengths that enlarge the toolkit beyond classical variety theory for nonassociative baric algebras.

minor comments (2)
  1. The abstract states that generators are obtained and applied in univariate and multivariate cases, but the manuscript would benefit from a short illustrative computation of one such generator (e.g., the simplest homogeneous case) already in the introduction to make the tree-labeling procedure immediately concrete.
  2. Notation for the algebraic-system operations on the free algebra generated by T is introduced without an accompanying small table or diagram listing the binary operations and their action on generators; adding such a table would improve readability of the subsequent closure-property arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit constructions

full rationale

The paper defines an algebraic system on the free commutative nonassociative algebra generated by T, uses rooted binary trees to compute Peirce polynomials explicitly, verifies that mutation algebras satisfy all Peirce-evanescent identities by direct computation, and concludes that arbitrary subsets of K arise as spectra. These steps are carried out by definition of operations and explicit enumeration of generators in univariate and multivariate cases, with no reduction of a claimed prediction to a fitted input, no self-citation load-bearing the central claim, and no ansatz or uniqueness imported from prior author work. The chain from axioms to spectrum statement remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper rests on the newly introduced algebraic-system structure and the definition of Peirce-evanescent identities; no external benchmarks or machine-checked results are mentioned.

axioms (1)
  • domain assumption The free commutative nonassociative algebra generated by a set T admits an algebraic system structure that endows classes of baric algebras satisfying a given set of identities with properties similar to varieties of algebras.
    This structure is the central device used to construct the identities and is invoked throughout the procedures described in the abstract.
invented entities (2)
  • Peirce-evanescent baric identity no independent evidence
    purpose: A polynomial identity verified by baric algebras whose associated Peirce polynomial is the zero polynomial.
    Newly defined concept whose properties are developed in the paper.
  • Peirce polynomial no independent evidence
    purpose: Polynomial attached to an identity that is represented via rooted binary trees with labeled leaves.
    Auxiliary object introduced to explain and manipulate the identities.

pith-pipeline@v0.9.0 · 5679 in / 1439 out tokens · 36368 ms · 2026-05-25T09:02:03.394230+00:00 · methodology

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