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arxiv: 2604.08446 · v1 · submitted 2026-04-09 · 🧮 math.LO

Probabilistic equational spectrum, primality and approximation in finite algebras

Carles Card\'o This is my paper

Pith reviewed 2026-05-10 17:22 UTC · model grok-4.3

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keywords probabilistic equational spectrumprimalityfinite algebrasequational probabilityfunctional approximationuniversal algebratwo-element algebras
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The pith

A probability measure on the equations of a finite algebra quantifies its primality and functional approximation ability.

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

By defining the probability of an equation as the proportion of tuples satisfying it, the paper constructs the probabilistic spectrum and studies its density and limit points in relation to primality. It introduces Prim(A) as a number in [0,1] that measures the algebra's capacity to approximate arbitrary functions through its operations. The work proves that this measure is at most 1/2 for every non-primal algebra with exactly two elements. This matters because it provides a uniform way to assess how close any finite algebra comes to generating all possible functions on its domain rather than treating primality as a binary property.

Core claim

The central discovery is that the set of probabilities obtained from equations in a finite algebra forms a spectrum whose properties are linked to primality, and that the associated Prim(A) value in the unit interval fully characterizes the algebra's ability to approximate all functions on its finite domain. In particular, the paper shows that non-primal two-element algebras always satisfy Prim(A) ≤ 1/2.

What carries the argument

The Prim(A) quantitative measure of primality derived from the probabilistic equational spectrum.

If this is right

  • The degree of primality is related to the size of the probabilistic spectrum.
  • Non-primal two-element algebras obey the bound Prim(A) ≤ 1/2.
  • The spectrum's structure, such as density and limit points, reflects the algebra's primality properties.
  • Prim(A) indicates the extent to which term operations can approximate any function on the domain.

Where Pith is reading between the lines

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

  • This approach suggests a way to rank algebras by their approximation power even when they are not primal.
  • Similar probabilistic spectra might be defined for other algebraic properties such as congruence distributivity.
  • Computational experiments on small algebras could verify the bound and explore its tightness for larger domains.

Load-bearing premise

The probabilistic definitions align with the classical notions of primality in universal algebra.

What would settle it

A counterexample would be any non-primal algebra on two elements for which the computed Prim(A) exceeds 1/2.

Figures

Figures reproduced from arXiv: 2604.08446 by Carles Card\'o.

Figure 1
Figure 1. Figure 1: The lattice Mn at the top of the figure and below a scheme of the orbits of M2 n/ Aut(Mn) from Example 3.6. The orbits are separated by lines, with the exception of the center of the table, where the gray cells form a single orbit of length n 2 − n, whereas the elements on the diagonal form an orbit of length n. An algebra is said to be automorphism-primal when the converse also holds; see [20]. We denote … view at source ↗
Figure 2
Figure 2. Figure 2: Quadrilateral in the proof of Lemma 6.1. The segments represent the distances according to the normalized Hamming metric. That is, α + 2¯ε ≥ µ(tα, ta) ≥ α − 2¯ε. Since µ(tα, ta) = Pr(tα ≈ ta | A), we obtain Pr(tα ≈ ta | A) ∈ [α − 2¯ε, α + 2¯ε]. It remains to remove the assumption that α is n-adic. This can be done directly, since n-adic numbers are dense, and for any real number we can find an n-adic numbe… view at source ↗
read the original abstract

We define the probability of an equation in a finite algebra as the proportion of tuples in its domain that satisfy it. We call the probabilistic spectrum of an algebra the set of probability values obtained when the equation varies. We study fundamental properties of this spectrum, such as density and limit points, and show that its structure is related to several notions of primality of an algebra. We introduce a quantitative measure of primality $\Prim(\A)\in[0,1]$ that characterizes the functional approximation capacity. We show that the degree of primality is related to the size of the spectrum. We also prove that all non-primal two-element algebras satisfy the universal bound $\Prim(\A)\le 1/2$.

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 / 3 minor

Summary. The paper claims to define the probability of an equation in a finite algebra A as the proportion of tuples in A^k satisfying the equation for suitable k. The probabilistic spectrum is the set of all such attainable probabilities. It studies density and limit points of the spectrum and relates its structure to primality. A quantitative measure Prim(A) ∈ [0,1] is introduced to characterize the functional approximation capacity of the algebra. The degree of primality is shown to relate to the size of the spectrum. Finally, it is proved that every non-primal two-element algebra satisfies the bound Prim(A) ≤ 1/2.

Significance. Should the results be verified, particularly the universal bound, this introduces a probabilistic lens on equational logic and primality in finite algebras. The measure Prim(A) offers a way to quantify how 'close' an algebra is to being primal in terms of approximation. The bound for two-element algebras is a specific, testable result that strengthens the connection between classical primality and the new probabilistic notions. This could be significant for researchers in universal algebra and clone theory.

major comments (1)
  1. [Proof of the bound Prim(A) ≤ 1/2 for non-primal two-element algebras] This result depends on exhaustive enumeration of all clones on the two-element set and computation of the probabilistic spectrum for each. The manuscript should provide a complete list or reference to the classification of clones on {0,1} and show explicitly that for non-primal clones, the maximum agreement probability over term pairs leads to Prim(A) ≤ 1/2. Gaps in this analysis, such as missing clones or incorrect normalization of probabilities across arities, would invalidate the universal bound.
minor comments (3)
  1. The notation for the probabilistic spectrum should be introduced with a formal definition, perhaps as Spec(A) = {p | p is the probability of some equation}.
  2. Add a table or list of examples for two-element algebras to support the bound claim.
  3. Review the paper for consistency in the use of 'primality' – distinguish classical from the new probabilistic version clearly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for recognizing the significance of introducing a probabilistic perspective on primality in finite algebras. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: [Proof of the bound Prim(A) ≤ 1/2 for non-primal two-element algebras] This result depends on exhaustive enumeration of all clones on the two-element set and computation of the probabilistic spectrum for each. The manuscript should provide a complete list or reference to the classification of clones on {0,1} and show explicitly that for non-primal clones, the maximum agreement probability over term pairs leads to Prim(A) ≤ 1/2. Gaps in this analysis, such as missing clones or incorrect normalization of probabilities across arities, would invalidate the universal bound.

    Authors: We agree with the referee that the proof of Prim(A) ≤ 1/2 for non-primal two-element algebras is based on the exhaustive classification of clones on {0,1} via Post's lattice. The manuscript references this classification implicitly through the known list of clones but does not provide an explicit list or table of computations for each non-primal clone. To address this, we will revise the manuscript to include a reference to Post's classification of clones and add an explicit verification, such as a table, showing the probabilistic spectrum and Prim(A) value for each relevant clone, confirming the bound holds with the correct normalization across arities. This will eliminate any potential gaps in the presentation. revision: yes

Circularity Check

0 steps flagged

No circularity; definitions precede independent theorem on two-element case

full rationale

The paper first defines the probability of an equation as the proportion of domain tuples satisfying it and the probabilistic spectrum as the resulting set of values. It then introduces Prim(A) as a new quantitative measure of primality tied to functional approximation capacity and shows its relation to spectrum size. The universal bound Prim(A) ≤ 1/2 for non-primal two-element algebras is stated as a derived theorem obtained by exhaustive enumeration over the finitely many clones on a two-element domain (Post's lattice), using the standard classical notion of primality for comparison. No step reduces a claimed prediction or result to a fitted parameter, self-citation, or definitional equivalence; the two-element proof is a standard finite case analysis independent of the new probabilistic constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Central claims rest on standard definitions of finite algebras, equations, and uniform probability over tuples; primality notions are imported from universal algebra without new axioms invented here.

axioms (2)
  • standard math Finite algebras are sets with finitary operations; equations are identities between terms.
    Invoked in the opening definitions of probability for equations.
  • domain assumption Probability is the uniform measure over all tuples in the finite domain.
    Used to define the probability of an equation holding.
invented entities (2)
  • Probabilistic spectrum no independent evidence
    purpose: Set of all probability values obtained by varying equations in the algebra.
    Newly defined object whose density and limit points are studied.
  • Prim(A) no independent evidence
    purpose: Quantitative measure in [0,1] of primality and functional approximation capacity.
    Introduced to characterize approximation and related to spectrum size.

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Forward citations

Cited by 1 Pith paper

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

  1. On semigroups and groupoids with minimal probabilistic spectrum

    math.LO 2026-03 unverdicted novelty 6.0

    Groupoids with minimal equational probabilistic spectrum are quasigroups apart from trivial cases, weak associativity conditions collapse to associativity, and semigroups with this spectrum are completely classified.

Reference graph

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