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arxiv: 2409.15541 · v3 · submitted 2024-09-23 · 🧮 math.RA · math.GR

On semigroups that are prime in the sense of Tarski, and groups prime in the senses of Tarski and of Rhodes

George M. Bergman This is my paper

Pith reviewed 2026-05-23 20:18 UTC · model grok-4.3

classification 🧮 math.RA math.GR
keywords semigroupsTarski primenessprime objectsmonoidsgroupsRhodes primenessdirect productsisomorphism classes
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The pith

The category of nonempty semigroups contains no objects that are prime according to Tarski's definition.

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

Tarski defined an object in a category closed under direct products to be prime if its isomorphism class divides a product in the monoid only by dividing one factor. The paper proves that no nonempty semigroup satisfies this. Examples of such primes appear in the monoid category and certain semigroup subcategories. Related primeness notions from Rhodes and others are compared in the group case.

Core claim

In the monoid of isomorphism classes of nonempty semigroups under direct product, no nonidentity element is prime in Tarski's sense, meaning every such class divides some product without dividing either factor. This contrasts with the category of monoids, where prime objects exist, and extends to comparisons with Rhodes primeness for groups.

What carries the argument

The monoid M_C of isomorphism classes under direct product, with Tarski primeness defined as dividing products only by dividing factors.

Load-bearing premise

That Tarski's definition of primeness applies directly and without modification to the monoid of isomorphism classes for the full category of all nonempty semigroups.

What would settle it

Exhibiting one nonempty semigroup whose isomorphism class divides a product of two others without dividing either factor would falsify the no-prime-objects claim.

read the original abstract

If $\mathcal{C}$ is a category of algebras closed under finite direct products, and $M_\mathcal{C}$ the commutative monoid of isomorphism classes of members of $\mathcal{C},$ with operation induced by direct product, A.Tarski defined a nonidentity element $p$ of $M_\mathcal{C}$ to be prime if, whenever it divides a product of two elements in that monoid, it divides one of them, and called an object of $\mathcal{C}$ prime if its isomorphism class has this property. McKenzie, McNulty and Taylor ask whether the category of nonempty semigroups has any prime objects. We show in section 2 that it does not. However, for the category of monoids, and some other subcategories of semigroups, we obtain examples of prime objects in sections 3-4. In section 5, two related questions open so far as I know, are recalled. In section 6, which can be read independently of the rest of this note, we recall two related conditions that are called primeness by semigroup theorists, and obtain results and examples on the relationships among those two conditions and Tarski's, in categories of groups. Section 7 notes an interesting characterization of one of those conditions when applied to finite algebras in an arbitrary variety. Various questions are raised.

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 studies Tarski primeness for objects in categories of algebras closed under finite direct products, where primeness is defined via the monoid of isomorphism classes under direct product. It proves that the category of all nonempty semigroups has no prime objects (Section 2), constructs examples of prime objects in the category of monoids and in certain proper subcategories of semigroups (Sections 3–4), recalls two open questions (Section 5), compares Tarski primeness with the notions of Rhodes and of another semigroup-theoretic condition in categories of groups (Section 6), and gives a characterization of one of those conditions for finite algebras in an arbitrary variety (Section 7).

Significance. The non-existence result in Section 2 supplies a direct negative answer to the question posed by McKenzie, McNulty and Taylor. The positive constructions in restricted categories and the explicit comparisons among three distinct primeness notions in groups clarify the scope of Tarski’s definition. Section 6 being readable independently and the variety-theoretic characterization in Section 7 are additional strengths.

minor comments (2)
  1. [Section 5] Section 5 states two open questions but does not indicate whether they are expected to be resolved within the same framework as the earlier sections.
  2. The abstract and introduction both cite McKenzie, McNulty and Taylor but the manuscript does not supply a bibliographic entry for their work.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its contributions (including the negative answer to the McKenzie-McNulty-Taylor question and the independent readability of Section 6), and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper applies Tarski's external definition of primeness verbatim to the monoid M_C of isomorphism classes under direct product (section 2), proving non-existence by explicit construction of factor pairs for any candidate non-unit. No parameters are fitted, no self-definitional loops appear in the monoid or division relation, and no load-bearing self-citations or ansatzes are invoked; all steps are direct from the stated definitions and standard category constructions. Sections 3-7 similarly derive examples and comparisons from the given notions without reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard definitions from Tarski and category theory; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Category C of algebras is closed under finite direct products, inducing the monoid M_C of isomorphism classes
    Invoked in the opening definition to set up the monoid operation and the notion of primeness.

pith-pipeline@v0.9.0 · 5782 in / 1091 out tokens · 29619 ms · 2026-05-23T20:18:06.550554+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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