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Left Ehresmann monoids with a proper basis

This paper develops a structural theory for left Ehresmann monoids by introducing the concept of a "proper basis," demonstrating that any such monoid with a proper basis is isomorphic to a specific subsemigroup $\mathcal{Q}_{\ell}(T,X,Y)$ , thereby establishing an analogue to the McAlister and O'Carroll theory for proper inverse semigroups.

Original authors: Gracinda Gomes, Victoria Gould, Yanhui Wang

Published 2026-04-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gracinda Gomes, Victoria Gould, Yanhui Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast library of mathematical structures called monoids. These are like rulebooks for combining things (like numbers, shapes, or even words) where the order of operations matters, but there's always a "do nothing" button (an identity element) that leaves things unchanged.

For decades, mathematicians have been fascinated by a special, well-behaved type of monoid called an Inverse Semigroup. Think of these as the "perfectly organized" members of the library. In these groups, every item has a unique "undo" button, and their internal structure is so tidy that they can be described as a simple mix of two things: a Group (a set of things that can be reversed) and a Semilattice (a set of things that can be compared and ordered, like a family tree).

This paper, titled "Left Ehresmann Monoids with a Proper Basis," tackles a messier, more chaotic cousin of these perfect groups: Left Ehresmann Monoids.

The Problem: The Messy Room

While the "perfect" groups have a neat "undo" button, Left Ehresmann Monoids are like a room where you can only undo things in one direction (left), and the rules for how things combine are much looser. They don't follow the strict "ample identity" that makes the perfect groups so easy to describe. Because of this, mathematicians couldn't easily build a "blueprint" for them. They knew these monoids existed and had a certain structure (called $P_\ell(T, X)$ ), but they lacked a way to describe exactly which ones were the "well-behaved" ones, similar to how the perfect groups were described.

The Solution: The "Proper Basis"

The authors introduce a new concept called a "Proper Basis."

Think of a Proper Basis as a special set of "building blocks" or "ingredients" for these monoids.

The Ingredients: Imagine you are building a tower. You have a pile of bricks (the monoid elements). A "Proper Basis" is a specific, curated selection of bricks that allows you to build any tower in the library in exactly one way.
The "Proper" Rule: The authors define a rule for these bricks: if two different stacks of bricks look the same from a distance (they belong to the same "congruence class") and they share the same "bottom brick" (a specific property called the $\ast$ -operation), then they must actually be the exact same stack. No duplicates, no confusion.

The Big Discovery: The "Q-Construction"

The paper's main achievement is proving that any Left Ehresmann Monoid that has this "Proper Basis" can be built using a specific recipe they call $Q_\ell(T, X, Y)$ .

Here is the analogy for this recipe:

The Stage ( $X$ ): Imagine a large, flat landscape (a semilattice) where you can walk around.
The Actors ( $T$ ): Imagine a group of actors (a monoid) who can walk around this landscape.
The Rules: The actors can move around, but they can only walk on certain paths, and they must follow specific rules about where they can go.
The Construction ( $Q_\ell$ ): The authors show that if you take these actors and restrict their movement to a specific, smaller, well-behaved part of the landscape (a sub-semilattice $Y$ ), you create a new structure.

They prove that every Left Ehresmann Monoid with a Proper Basis is essentially just one of these restricted structures. It's like saying, "Every well-behaved messy room is actually just a specific type of organized apartment."

Why This Matters (In Simple Terms)

Before this paper, mathematicians had a general description for these messy monoids ( $P_\ell(T, X)$ ), but it was too broad. It was like having a map of the entire world, but needing a map of just the city.

This paper provides the "city map." It identifies the specific subset of these monoids that behave nicely (those with a Proper Basis) and shows that they are structurally identical to the "restricted apartment" model ( $Q_\ell$ ).

The "Globalization" Side Quest

To prove this, the authors had to solve a side puzzle involving Partial Actions.

Imagine a dance where some dancers only know how to dance with certain partners, and only on certain parts of the floor. This is a "partial action."
The authors proved that you can always expand this partial dance into a full dance where everyone knows how to dance with everyone, as long as the original rules were followed correctly. They called this "Globalization." This mathematical trick was essential to build their "city map."

The Bottom Line

The authors have successfully built a theory for Left Ehresmann Monoids that mirrors the famous theory for Inverse Semigroups. They found the "Proper Basis" (the secret ingredient list) and showed that any monoid with this ingredient list is isomorphic (structurally identical) to a specific, well-defined construction ( $Q_\ell$ ).

They also note that while they focused on monoids (which have an identity element), the same ideas might apply to semigroups (which don't), but that is a question for future research. They also briefly mention that their approach is different from another recent paper by Kudryavtseva and Laan, and they invite others to explore the deeper connections between these two different ways of looking at the same mathematical objects.

1. Problem Statement and Motivation

Context:
The paper addresses the structural theory of left Ehresmann monoids. These are unary monoids satisfying specific identities that generalize inverse semigroups and left restriction semigroups. While the theory of inverse semigroups is well-established (notably via McAlister's $P$ -semigroups and semidirect products of groups and semilattices), the theory for left Ehresmann monoids has lagged behind.

The Gap:

Inverse Semigroups: Every inverse semigroup has a "cover" (an idempotent-sepreating preimage) that is a semidirect product of a semilattice by a group. Furthermore, "proper" inverse semigroups (those embeddable in such products) are characterized by McAlister's $P$ -semigroups.
Left Ehresmann Monoids: It was known that any left Ehresmann monoid has a cover of the form $P_\ell(T, X)$ , constructed from a monoid $T$ acting on a semilattice $X$ via order-preserving maps. However, there was no analogue to the concept of a "proper" inverse semigroup or a $P$ -semigroup for this class.
The Challenge: Unlike inverse or restriction semigroups, left Ehresmann monoids do not satisfy the "ample identity" ( $xy^+ = (xy)^+x$ ). This failure prevents the standard decomposition techniques used in inverse semigroup theory (which rely on rearranging idempotents) from working directly. Consequently, free left Ehresmann monoids are described using labelled trees rather than Cayley graphs, and their structure is more complex.

Objective:
The authors aim to fill this theoretical gap by:

Defining a notion of a proper basis for left Ehresmann monoids.
Proving that monoids with a proper basis possess structural properties analogous to proper inverse semigroups.
Establishing a structure theorem characterizing these monoids as subsemigroups of $P_\ell(T, X)$ , analogous to McAlister's $P$ -semigroups.

2. Methodology and Key Concepts

The authors develop a new framework by introducing specific algebraic structures and leveraging a "globalisation" result for partial actions.

A. $\ast$ -Left Ehresmann Monoids

To handle the lack of the ample identity, the authors introduce a second unary operation, denoted by $\ast$ , alongside the standard left operation $+$ .

A $\ast$ -left Ehresmann monoid is a left Ehresmann monoid equipped with a unary operation $\ast$ such that $E = \{a^+ : a \in M\} = \{a^\ast : a \in M\}$ is a semilattice, and $a^\ast$ is the least right identity for $a$ from $E$ .
They prove that any left Ehresmann monoid with uniqueness of $T$ -normal forms (a decomposition $a = t_0 e_1 t_1 \dots e_n t_n$ ) naturally admits this second operation, making it $\ast$ -left Ehresmann.

B. Proper Basis and Atomic Subsets

The core innovation is the definition of a proper basis.

$\ast$ -subset: A subset $H \subseteq M$ where every $h \in H$ has a minimal right identity $h^\ast \in E$ .
Atomic: $H$ satisfies specific closure properties (H1–H5) regarding products and congruences.
Proper: $H$ is proper if $h, k \in H$ with $h^\ast = k^\ast$ and $h \sigma k$ (where $\sigma$ is the least congruence identifying projections) implies $h = k$ .
Basis: $H$ is a basis if it generates $M$ and every element has a unique $H$ -canonical form (analogous to $T$ -normal forms but defined via the basis elements).

C. Globalisation of Partial Actions

A crucial technical tool is the extension of the globalisation theorem for partial actions.

The authors prove that a strong, order-preserving partial action of a monoid $T$ on a semilattice $Y$ can be globalised to a full, order-preserving action on a larger semilattice $X$ .
This result (Theorem 4.17) is essential for constructing the ambient monoid $P_\ell(T, X)$ from a given left Ehresmann monoid.

D. The Monoids $P_\ell(T, X)$ and $Q_\ell(T, X, Y)$

$P_\ell(T, X)$ : The "semidirect product" analogue for left Ehresmann monoids. It is constructed from a monoid $T$ and a semilattice $X$ with an order-preserving action. It has a proper basis $H_P = \{te : t \in T, e \in X\}$ .
$Q_\ell(T, X, Y)$ : A subsemigroup of $P_\ell(T, X)$ generated by $H_Q = \{te \in H_P : e \in Y \text{ and } t \cdot e \in Y\}$ , where $Y$ is a subsemilattice of $X$ satisfying specific conditions (A and B). These play the role of McAlister's $P$ -semigroups.

3. Key Contributions and Results

1. Introduction of the Proper Basis

The paper defines the concept of a proper basis for left Ehresmann monoids. This concept captures the abstract notion of "properness" (analogous to $E$ -unitary inverse semigroups) in a setting where the ample identity fails.

2. Structure of Monoids with a Proper Basis

The authors prove that any left Ehresmann monoid with a proper basis is necessarily $\ast$ -left Ehresmann. This allows them to utilize the second unary operation $\ast$ to define unique canonical forms and analyze the structure of the monoid.

3. The Main Structure Theorem (Theorem 8.1)

The central result of the paper establishes a structural characterization:

Theorem: A biunary monoid $Q$ is a left Ehresmann monoid with a proper basis if and only if $Q$ is isomorphic to a biunary monoid subsemigroup $Q_\ell(T, X, Y)$ of some $P_\ell(T, X)$ .

This theorem mirrors the classic result for inverse semigroups (where proper inverse semigroups are isomorphic to $P$ -semigroups).

$P_\ell(T, X)$ plays the role of the semidirect product $X \rtimes G$ .
$Q_\ell(T, X, Y)$ plays the role of the $P$ -semigroup.
The isomorphism is constructed by mapping the proper basis of $Q$ to the generating set of $Q_\ell$ .

4. Existence of Proper Bases

The authors demonstrate that:

The monoids $P_\ell(T, X)$ themselves possess a proper basis ( $H_P$ ).
The submonoids $Q_\ell(T, X, Y)$ possess a proper basis ( $H_Q$ ).
Free left Ehresmann monoids possess a proper basis, implying they are isomorphic to some $Q_\ell(T, X, Y)$ .

5. Globalisation Result

The paper provides a rigorous proof that a strong, order-preserving partial action of a monoid on a semilattice can be globalised to a full action on a semilattice. This result is of independent interest in semigroup theory and order theory.

4. Significance and Impact

Paradigm Shift: The paper successfully adapts the structural theory of inverse semigroups to the more complex class of left Ehresmann monoids, despite the absence of the ample identity. It moves the field away from reliance on labelled trees (used for free objects) toward a decomposition theory based on semilattices and monoid actions.
Unification: It unifies the study of free left Ehresmann monoids, restriction monoids, and general left Ehresmann monoids under a single structural framework ( $Q_\ell(T, X, Y)$ ).
New Tools: The introduction of the $\ast$ -operation and the concept of a proper basis provides new algebraic tools for analyzing non-regular semigroups.
Future Directions: The authors explicitly outline open questions regarding the extension of this theory to semigroups (without identity) and the relationship between their "proper basis" approach and the category-theoretic approach to proper Ehresmann semigroups developed by Kudryavtseva and Laan.

In summary, this paper provides the foundational "McAlister-type" structure theorem for left Ehresmann monoids, establishing that those with a proper basis are structurally equivalent to specific subsemigroups of order-preserving semidirect products, thereby completing a major chapter in the theory of semigroups with a semilattice of idempotents.