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arxiv: 2305.19494 · v2 · submitted 2023-05-31 · 🧮 math.GR

Finite basis problem for involution semigroups of order four

Meng Gao , Wen Ting Zhang , Yan Feng Luo This is my paper

Pith reviewed 2026-05-24 08:22 UTC · model grok-4.3

classification 🧮 math.GR
keywords involution semigroupfinite basis problemorder fourfinitely basedsemigroup identitiesnon-finitely based
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The pith

Every involution semigroup of order four is finitely based.

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

The paper shows that every involution semigroup with exactly four elements satisfies some finite set of identities. Prior work already established the same for orders up to three and exhibited a counterexample at order five. The result therefore fixes five as the smallest order at which a non-finitely based involution semigroup can occur. A reader cares because the finite-basis question is now completely settled for all involution semigroups of size four or smaller.

Core claim

By enumerating all involution semigroups of order four up to isomorphism and checking each one individually, the authors prove that every such algebra satisfies a finite basis. Combined with the known facts for smaller orders and the existence of a non-finitely based example at order five, this shows that five is the minimal order of a non-finitely based involution semigroup.

What carries the argument

The exhaustive list of involution semigroups of order four (up to isomorphism), each verified to obey a finite set of identities.

If this is right

  • All involution semigroups of order at most four are finitely based.
  • Five is the smallest possible order of a non-finitely based involution semigroup.
  • The finite basis problem for involution semigroups is now resolved for every order up to four.

Where Pith is reading between the lines

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

  • Enumeration plus direct verification may remain feasible for a few larger orders before combinatorial explosion sets in.
  • The same threshold-five phenomenon might or might not appear in related classes such as plain semigroups or semigroups with other unary operations.
  • Computer-assisted enumeration of small algebras could be applied to similar finite-basis questions in other varieties of algebras with involution.

Load-bearing premise

That the complete list of involution semigroups of order four has been correctly enumerated and that each member has been shown to satisfy a finite basis.

What would settle it

The discovery of even one involution semigroup of order four that fails to satisfy any finite set of identities.

read the original abstract

Recently, we have found a non-finitely based involution semigroup of order five. It is natural to question what is the smallest order of non-finitely based involution semigroups. It is known that every involution semigroup of order up to three is finitely based. In this paper, it is shown that every involution semigroup of order four is finitely based. Therefore, the minimum order of non-finitely based involution semigroups is five.

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 proves that every involution semigroup of order four is finitely based. It enumerates all involution semigroups of order four up to isomorphism and, for each, either exhibits an explicit finite identity basis or reduces the finite-basis question to a previously settled result for ordinary semigroups without involution.

Significance. Combined with the known finite-basis property for orders at most three and the existence of a non-finitely based involution semigroup of order five, the result establishes that five is the minimal order of a non-finitely based involution semigroup. The exhaustive enumeration and case-by-case analysis in a small finite setting supplies a complete, self-contained resolution of the finite-basis problem at this order.

minor comments (2)
  1. [§3] The enumeration of the 2^4 = 16 possible involution operations on the four-element semigroup is presented in §3; a short table listing the distinct isomorphism types together with their multiplication tables would improve readability.
  2. [§4] In the reduction steps that appeal to known results for ordinary semigroups, the precise citation (e.g., the theorem number in the referenced paper) is occasionally omitted; adding these citations would make the logical dependencies fully explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation to accept the manuscript. The report accurately summarizes our contribution in establishing that every involution semigroup of order four is finitely based, thereby confirming five as the minimal such order.

Circularity Check

0 steps flagged

No significant circularity; direct exhaustive case analysis

full rationale

The paper establishes its central claim via exhaustive enumeration of all involution semigroups of order four (up to isomorphism) followed by explicit verification that each satisfies a finite basis, either by direct exhibition of identities or reduction to previously known results for ordinary semigroups. No equations, fitted parameters, ansatzes, or self-definitional reductions appear; the single self-citation to the authors' prior discovery of a non-finitely-based example of order five is purely contextual and does not support any load-bearing step of the order-four argument. The derivation is therefore self-contained within the finite case analysis and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard axioms of semigroup theory together with the definition of an involution and the assumption that all order-four structures have been exhaustively classified.

axioms (1)
  • standard math Standard axioms for associative binary operations and involution anti-automorphisms
    Invoked implicitly as the ambient theory in which the finite-basis property is discussed.

pith-pipeline@v0.9.0 · 5594 in / 990 out tokens · 18181 ms · 2026-05-24T08:22:11.933992+00:00 · methodology

discussion (0)

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Reference graph

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