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arxiv: 2605.15493 · v1 · pith:ONYACONCnew · submitted 2026-05-15 · 🧮 math.GR

A nonfinitely based additively idempotent semiring of order four

Mengya Yue , Miaomiao Ren This is my paper

Pith reviewed 2026-05-19 15:52 UTC · model grok-4.3

classification 🧮 math.GR
keywords additively idempotent semiringnonfinitely basedfinite basisidentitiessemiring of order fouradditive reduct
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The pith

A 4-element additively idempotent semiring has no finite basis for its identities.

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

The paper first establishes a sufficient condition under which an additively idempotent semiring fails to have a finite basis for its identities. It then applies the condition to produce several concrete examples, including the 4-element semiring S_{(4,124)} whose additive reduct contains two minimal elements and two coatoms. The result is that the identities of these semirings cannot be derived from any finite subset. Readers care because finite bases make equational theories easier to work with computationally and structurally in algebra.

Core claim

The authors prove a sufficient condition that forces an additively idempotent semiring to be nonfinitely based. They verify the condition on multiple examples, notably the semiring S_{(4,124)} of order four whose additive reduct has two minimal elements and two coatoms. It follows that these semirings have no finite basis for their identities.

What carries the argument

The sufficient condition for an additively idempotent semiring to be nonfinitely based, verified by checking structural features of its additive reduct.

If this is right

  • The 4-element semiring S_{(4,124)} has no finite basis for its identities.
  • Other additively idempotent semirings that meet the sufficient condition are likewise nonfinitely based.
  • Nonfinite basis can occur among additively idempotent semirings of very small finite order.

Where Pith is reading between the lines

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

  • The condition may help locate additional small nonfinitely based examples in this class of algebras.
  • Analogous conditions could be sought for related structures such as semigroups or lattices.
  • Classifying additively idempotent semirings by order according to whether they admit finite bases becomes a natural next question.

Load-bearing premise

The sufficient condition is valid and correctly holds for the 4-element semiring whose additive reduct has two minimal elements and two coatoms.

What would settle it

Deriving every identity of S_{(4,124)} from some finite list of identities would show that the semiring is in fact finitely based.

read the original abstract

We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element semiring $S_{(4,124)}$ whose additive reduct has two minimal elements and two coatoms. Consequently, these semirings have no finite basis for their identities.

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 paper first establishes a sufficient condition for an additively idempotent semiring to be nonfinitely based. It then applies the condition to several examples, including the 4-element semiring S_{(4,124)} whose additive reduct has two minimal elements and two coatoms, and concludes that these semirings have no finite basis for their identities.

Significance. If the sufficient condition holds and is correctly verified on the given examples, the work supplies concrete small-order instances of nonfinitely based additively idempotent semirings. The explicit 4-element example is noteworthy for its minimality and may assist in mapping the boundary between finitely based and nonfinitely based varieties in this class of algebras.

minor comments (2)
  1. The abstract states that 'several examples' are exhibited but does not indicate their number or the orders involved beyond the 4-element case; a brief enumeration would improve clarity.
  2. Notation S_{(4,124)} is introduced without an immediate definition of the subscript; a short parenthetical explanation or forward reference to the construction in Section 3 would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of the significance of the 4-element example, and the 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

full rationale

The paper first proves a general sufficient condition for an additively idempotent semiring to lack a finite equational basis, then verifies that the specific 4-element semiring S_{(4,124)} (whose additive reduct has two minimal elements and two coatoms) satisfies the hypotheses of that condition. This is a standard theorem-then-application structure with no reduction of the central claim to a fitted parameter, self-definition, or load-bearing self-citation. The verification for the small finite algebra is direct and independent of the result being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract indicates reliance on standard algebraic definitions and a newly proved sufficient condition; no free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (1)
  • standard math Basic definitions and properties of semirings and idempotent addition
    The paper presupposes the standard axioms of semiring theory and additive idempotence.

pith-pipeline@v0.9.0 · 5583 in / 1211 out tokens · 52997 ms · 2026-05-19T15:52:10.512044+00:00 · methodology

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Reference graph

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