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arxiv: 2604.07152 · v1 · submitted 2026-04-08 · 🧮 math.CT

Recognition: no theorem link

Embedding Boolean ample monoids as full submonoids of Boolean inverse monoids

Mark V. Lawson

Pith reviewed 2026-05-10 18:14 UTC · model grok-4.3

classification 🧮 math.CT
keywords Boolean ample monoidsBoolean inverse monoidsfull submonoidsgroupoids of fractionsnon-commutative Stone dualityembeddingsinverse semigroupsmonoid embeddings
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The pith

Boolean ample monoids can be fully embedded as submonoids of Boolean inverse monoids under certain conditions, generalizing the embedding of right reversible cancellative monoids into groups.

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

The paper establishes that Boolean ample monoids may be embedded fully into Boolean inverse monoids in certain circumstances. This construction extends the classical case where right reversible cancellative monoids embed into groups. The argument proceeds by constructing groupoids of fractions and then applying non-commutative Stone duality to obtain the target inverse monoid. A reader would care because the result supplies a concrete bridge between two families of algebraic structures that appear in semigroup theory and inverse algebra.

Core claim

We show that, in certain circumstances, a Boolean ample monoid may be fully embedded into a Boolean inverse monoid in a way that generalizes how right reversible cancellative monoids may be embedded into groups. We use groupoids of fractions and non-commutative Stone duality to prove the result.

What carries the argument

Groupoids of fractions together with non-commutative Stone duality, which construct the containing Boolean inverse monoid and verify that the original monoid sits as a full submonoid.

If this is right

  • The Boolean ample monoid inherits algebraic and order-theoretic properties from its ambient Boolean inverse monoid.
  • Results proved for Boolean inverse monoids can be specialized back to the embedded ample monoid.
  • The embedding supplies a uniform way to associate an inverse monoid to an ample monoid whenever the circumstances hold.

Where Pith is reading between the lines

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

  • The same duality techniques might adapt to embed other restricted classes of monoids into inverse structures.
  • One could test the construction on concrete examples such as free ample monoids or monoids arising from partial actions to see the embedding explicitly.
  • The result hints at a possible dictionary between ample monoids and certain inverse monoids that could be explored through representation theory.

Load-bearing premise

The monoid satisfies the unspecified certain circumstances that permit the groupoid of fractions to form and the duality to produce a full embedding.

What would settle it

An explicit Boolean ample monoid that cannot be realized as a full submonoid of any Boolean inverse monoid would refute the claim.

read the original abstract

We show that, in certain circumstances, a Boolean ample monoid may be fully embedded into a Boolean inverse monoid in a way that generalizes how right reversible cancellative monoids may be embedded into groups. We use groupoids of fractions and non-commutative Stone duality to prove the result.

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 / 3 minor

Summary. The paper shows that Boolean ample monoids admitting a groupoid of fractions (i.e., satisfying the relevant Ore-type conditions) can be fully embedded as submonoids of Boolean inverse monoids. The construction first forms the groupoid of fractions of the monoid and then applies non-commutative Stone duality to obtain the Boolean inverse monoid, recovering the original monoid as the full submonoid consisting of the compact elements. This is presented as a generalization of the embedding of right reversible cancellative monoids into groups.

Significance. If the central construction holds, the result strengthens the links between ample monoids, inverse monoids, and non-commutative Stone duality, providing a systematic way to realize certain Boolean ample monoids inside Boolean inverse monoids. The explicit use of groupoids of fractions and the duality functor supplies a concrete, reusable method that may be applied to other classes of monoids satisfying analogous Ore conditions.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'in certain circumstances' is imprecise; while the body states the condition as the existence of a groupoid of fractions satisfying the appropriate Ore-type conditions, a concise restatement of this hypothesis in the abstract would improve readability.
  2. [§2] §2 (or the section introducing the groupoid of fractions): the verification that the embedding is full (i.e., that every morphism in the image arises from the original monoid) would benefit from an explicit diagram or a short lemma isolating the fullness property.
  3. [§3] The notation for the compact elements in the dual Boolean inverse monoid is introduced without a dedicated symbol; adding a consistent notation (e.g., K(M) or similar) would clarify the identification throughout the later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing the significance of the embedding result via groupoids of fractions and non-commutative Stone duality, and for recommending minor revision. Since no specific major comments were raised, we have no points to address point-by-point at this stage.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained via established external tools

full rationale

The paper constructs the embedding explicitly: form the groupoid of fractions of the Boolean ample monoid (under the stated Ore-type conditions), then apply non-commutative Stone duality to obtain the Boolean inverse monoid, recovering the original monoid as the submonoid of compact elements. This chain relies on prior independent results on ample monoids, groupoids of fractions, and Stone duality for inverse monoids, none of which are defined in terms of the target embedding or fitted to the present data. No equations reduce to self-definition, no parameters are fitted and relabeled as predictions, and self-citations (if present) are not load-bearing for the central claim. The result is a standard categorical construction generalizing the cancellative-to-group case and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of monoids, inverse monoids, and Boolean algebras, plus the applicability of groupoids of fractions and non-commutative Stone duality, which are treated as established background.

axioms (2)
  • domain assumption Boolean ample monoids satisfy the axioms of ample monoids with Boolean semilattice of idempotents.
    Invoked implicitly as the starting objects for the embedding.
  • domain assumption Groupoids of fractions exist for the monoids under the given circumstances.
    Central to the proof strategy mentioned in the abstract.

pith-pipeline@v0.9.0 · 5326 in / 1286 out tokens · 27864 ms · 2026-05-10T18:14:02.568422+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages

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