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arxiv: 2601.22923 · v2 · submitted 2026-01-30 · 🧮 math.RA

Left Ehresmann monoids with a proper basis

Gracinda Gomes , Victoria Gould , Yanhui Wang This is my paper

Pith reviewed 2026-05-16 09:37 UTC · model grok-4.3

classification 🧮 math.RA MSC 20M1820M10
keywords left Ehresmann monoidsproper basiscoversP-semigroupsorder-preserving actionssemilatticesbiunary monoids
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The pith

Any left Ehresmann monoid with a proper basis is isomorphic to a biunary subsemigroup Q_ℓ(T,X,Y) inside the cover P_ℓ(T,X).

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

The paper establishes a representation theorem for left Ehresmann monoids that possess a proper basis. It begins from the known fact that every such monoid admits a cover of the form P_ℓ(T,X), built from a monoid T acting by order-preserving maps on a semilattice X with identity. The proper basis condition is then shown to be preserved under passage to certain biunary subsemigroups Q_ℓ(T,X,Y) of these covers. The central result is that the presence of a proper basis forces the monoid to be isomorphic to one of these Q constructions. This mirrors the McAlister-O'Carroll theorem for proper inverse semigroups, supplying an explicit algebraic model in terms of actions on semilattices.

Core claim

Every left Ehresmann monoid that admits a proper basis is isomorphic to a biunary monoid subsemigroup Q_ℓ(T,X,Y) of some P_ℓ(T,X), where P_ℓ(T,X) is the left Ehresmann monoid obtained from a monoid T and an order-preserving action of T on a semilattice X with identity, and Q_ℓ(T,X,Y) is formed by restricting to a suitable subset Y of X that is closed under the relevant operations.

What carries the argument

The proper basis, which is a generating set satisfying specific projection and order conditions that allow the monoid to be recovered from the action data in Q_ℓ(T,X,Y).

If this is right

  • Left Ehresmann monoids with proper bases inherit structural features previously known only for two-sided Ehresmann monoids.
  • Every free left Ehresmann monoid possesses a proper basis and therefore admits the Q_ℓ representation.
  • Order-preserving partial actions of monoids on partially ordered sets or semilattices can be globalised to total actions.
  • The constructions P_ℓ(T,X) serve as universal covers in the same way semidirect products do for inverse semigroups.

Where Pith is reading between the lines

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

  • The result supplies a uniform way to test whether a given left Ehresmann monoid is proper by checking for the existence of a basis with the stated projection properties.
  • The globalisation theorem for partial actions may apply directly to other classes of monoids that act on posets, such as restriction semigroups.
  • One could ask whether the same Q_ℓ form classifies all left Ehresmann monoids once a suitable notion of basis is weakened.

Load-bearing premise

That the known cover of any left Ehresmann monoid can always be written in the concrete form P_ℓ(T,X) and that the single extra condition of possessing a proper basis is enough to guarantee an isomorphism to some Q_ℓ(T,X,Y) without further restrictions on the action.

What would settle it

A concrete left Ehresmann monoid equipped with a proper basis whose multiplication table or projection structure fails to match that of every possible Q_ℓ(T,X,Y) for any choice of T, X and Y.

read the original abstract

Left Ehresmann monoids, and their two-sided counterpart of Ehresmann monoids, were so named by Lawson, who elucidated their connection to the work of Ehresmann in differential geometry. This article is dedicated to building a theory for left Ehresmann monoids inspired by that for inverse semigroups; in order to do so we must develop substantially different ideas and techniques. It is known that every left Ehresmann monoid has a cover, that is, a projection separating preimage, of the form $\mathcal{P}_{\ell}(T,X)$, where $\mathcal{P}_{\ell}(T,X)$ is a left Ehresmann monoid constructed from a monoid $T$ and an order-preserving action of $T$ on a semilattice $X$ with identity. We introduce the notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$, and consequently any free left Ehresmann monoid, possesses a proper basis. We show that any left Ehresmann monoid with a proper basis displays properties close to those of two-sided Ehresmann monoids. Next, we exhibit a class of subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ (properly, biunary monoid subsemigroups) of the monoids $\mathcal{P}_{\ell}(T,X)$, which are also left Ehresmann with a proper basis. We prove that any left Ehresmann monoid with a proper basis is isomorphic to some $\mathcal{Q}_{\ell}(T,X,Y)$. Our results can be regarded as being analogous to those for proper inverse semigroups, due to McAlister and O'Carroll, the $\mathcal{Q}_{\ell}(T,X,Y)$ playing the role of the $P$-semigroups and the $\mathcal{P}_{\ell}(T,X)$ the role of the semidirect products of a semilattice by a group. In the process of proving our main theorems we present a globalisation result for an order-preserving partial action of a monoid on a partially ordered set or semilattice.

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 develops a representation theory for left Ehresmann monoids analogous to that for inverse semigroups. It introduces the notion of a proper basis, proves that every left Ehresmann monoid embeds into a cover of the form P_ℓ(T,X) (constructed from a monoid T acting order-preservingly on a semilattice X), shows that P_ℓ(T,X) and free left Ehresmann monoids possess proper bases, defines a class of biunary subsemigroups Q_ℓ(T,X,Y) that inherit proper bases, and proves that every left Ehresmann monoid with a proper basis is isomorphic to some Q_ℓ(T,X,Y). A globalisation result for order-preserving partial actions on posets or semilattices is established as a technical tool.

Significance. If the main isomorphism theorem holds, the work supplies a McAlister–O’Carroll-style structure theorem for left Ehresmann monoids with proper basis, with Q_ℓ(T,X,Y) playing the role of P-semigroups. This furnishes concrete, constructible models for an important subclass of left Ehresmann monoids and may support further development of their theory, including connections to Ehresmann’s original geometric ideas. The globalisation result for partial actions is a reusable technical contribution independent of the main theorem.

minor comments (3)
  1. The term 'biunary monoid subsemigroup' is used in the abstract and §4 without an explicit definition at first occurrence; a short inline definition or forward reference to the precise axioms would improve readability.
  2. In the construction of P_ℓ(T,X) (around §2–3), the order-preserving action is described via a map T × X → X, but the compatibility conditions with the semilattice operations are stated only implicitly; spelling out the two or three required identities would prevent misreading.
  3. The globalisation theorem (stated in §5) distinguishes partial actions on posets from those on semilattices; a single clarifying sentence on whether the semilattice case requires extra idempotent-preservation would remove a minor ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. We are pleased that the analogy to McAlister–O’Carroll theory and the independent value of the globalisation result for partial actions have been noted.

Circularity Check

0 steps flagged

No significant circularity; derivation builds from external cover theorem and new definitions

full rationale

The central structure theorem invokes the known external cover result that every left Ehresmann monoid embeds into a P_ℓ(T,X) constructed from a monoid action on a semilattice. The paper then introduces the new notion of proper basis, verifies it holds for all such P_ℓ(T,X) and for free left Ehresmann monoids, defines the class Q_ℓ(T,X,Y) explicitly as biunary subsemigroups inheriting a proper basis, and constructs the isomorphism using a globalisation result for partial actions that is proved within the paper. No equation or step reduces the target isomorphism to a fitted parameter, a self-citation chain, or a renaming of prior results; all objects are defined from standard monoids, semilattices, and order-preserving actions without self-definition. The McAlister–O’Carroll analogy is presented as motivation only.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The results rest on standard algebraic definitions of monoids and semilattices together with newly introduced notions of proper basis and the specific constructions P and Q; no numerical parameters are fitted.

axioms (2)
  • standard math A monoid is a set with associative binary operation and identity element
    Used as the base structure for T and the overall monoids.
  • standard math A semilattice is a commutative idempotent semigroup, equivalently a poset with meets
    Defines the structure X on which T acts.
invented entities (3)
  • proper basis no independent evidence
    purpose: Additional property ensuring left Ehresmann monoids behave like two-sided ones
    Newly defined to select the class of monoids for which the representation holds.
  • P_ℓ(T,X) no independent evidence
    purpose: Cover monoid constructed from monoid T and semilattice X with order-preserving action
    Explicitly built to serve as a universal cover for any left Ehresmann monoid.
  • Q_ℓ(T,X,Y) no independent evidence
    purpose: Biunary subsemigroup of P_ℓ(T,X) that is left Ehresmann with proper basis
    New family introduced to provide the concrete isomorphic copies.

pith-pipeline@v0.9.0 · 5695 in / 1519 out tokens · 32117 ms · 2026-05-16T09:37:52.124733+00:00 · methodology

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Reference graph

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