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Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids

This paper establishes general presentations for maximal subgroups of free projection- and idempotent-generated semigroups and applies them to partition monoids, revealing that while the former yield symmetric groups $S_r$ , the latter produce direct products $Z \times S_r$ due to a connection with twisted partition monoids.

Original authors: James East, Robert D. Gray, P. A. Azeef Muhammed, Nik Ruskuc

Published 2026-07-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: James East, Robert D. Gray, P. A. Azeef Muhammed, Nik Ruskuc

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 world built entirely out of shapes that can snap together, slide past one another, and sometimes stick in place forever. In the realm of mathematics, this is the universe of semigroups. Think of a semigroup as a giant, chaotic dance floor where every dancer (an element) has a specific move they can do. Some dancers are special: if they do their move twice, they end up exactly where they started. These are called idempotents. They are like the "pause" buttons of the dance floor; once you hit them, the action freezes.

Now, imagine you want to build the ultimate, most flexible dance floor possible using only these "pause" dancers. You don't want to add any extra moves or rules that aren't strictly necessary. This creates a "free idempotent-generated semigroup." It's a mathematical construction that asks: "If I only have these freeze-buttons, what kind of complex structures can I build?"

But there's a twist. Some of these dance floors have a special mirror property. If you look at a dancer in the mirror, they do a move that undoes the original one. This is called an involution, and it turns the dance floor into a "regular ∗-semigroup." In this mirrored world, the "pause" dancers have a simpler, more rigid cousin called a projection. The big question mathematicians have been asking is: "If we build our dance floor using only the rigid projections, does it look the same as the one built with the flexible idempotents? And what happens to the groups of dancers who get stuck in a loop, spinning in perfect circles?" These spinning loops are called maximal subgroups, and figuring out their shape is like discovering the secret DNA of the dance floor.

This paper is a deep dive into that very question, specifically looking at a famous dance floor called the partition monoid (denoted as $P_n$ ). The authors, James East, Robert D. Gray, P.A. Azeef Muhammed, and Nik Ruškuc, set out to compare two different ways of building this structure: one using the flexible "pause" buttons (idempotents) and one using the rigid "mirror" buttons (projections).

They found that while the two dance floors look almost identical from a distance, they have a secret, fundamental difference in how their dancers spin.

When they built the floor using the rigid projections (the free projection-generated semigroup, $PG(P)$), the spinning loops turned out to be exactly what you'd expect: the symmetric group $S_r$ . In plain English, this is just the set of all possible ways to rearrange $r$ items. If you have 3 dancers, there are 6 ways they can swap places. The math here is clean, tidy, and matches the original partition monoid perfectly.

However, when they built the floor using the flexible idempotents (the free idempotent-generated semigroup, $IG(E)$), something strange happened. The spinning loops weren't just the rearrangements anymore. They were the rearrangements plus an infinite, endless clock ticking in the background. Mathematically, this is the direct product of the integers ( $\mathbb{Z}$ ) and the symmetric group ( $S_r$ ).

The authors explain this "extra" infinite part by connecting it to a mysterious, twisted version of the partition monoid called the twisted partition monoid ( $P^\Phi_n$ ). Imagine that every time two dancers collide on this new floor, a little counter in the middle of the room ticks up or down depending on how many "floating" pieces of the dance floor are left in the air. This counter never stops; it can go up to infinity or down to negative infinity. The paper proves that the "free" version of the idempotent dance floor naturally inherits this ticking clock, even though the original partition monoid $P_n$ completely forgets about it.

The authors are very sure about these results. They didn't just guess or simulate; they constructed rigorous mathematical proofs using a technique called Reidemeister–Schreier rewriting. This is like taking a complex knot, labeling every strand, and systematically untangling it to reveal the exact shape of the loops inside. They showed that for any rank $r$ between 0 and $n-2$ , the projection-based group is exactly $S_r$ , while the idempotent-based group is definitely $\mathbb{Z} \times S_r$ .

They also ruled out the idea that these two structures are the same. In some other mathematical worlds (like the Temperley–Lieb monoid), the projection version and the idempotent version are identical twins. But for the partition monoid, the authors proved they are distinct cousins. The "free" idempotent version is strictly "larger" and more complex because of that infinite cyclic factor $\mathbb{Z}$ .

In the end, the paper solves a long-standing puzzle: it tells us exactly what the "spinning loops" look like in these free mathematical structures. It reveals that the simple act of choosing to build with "rigid mirrors" versus "flexible pauses" changes the fundamental nature of the group, adding an infinite dimension of time to the mix. This discovery doesn't just tidy up a math problem; it suggests that the "twist" in the twisted partition monoid is actually hidden inside the very fabric of the idempotent structure, waiting to be discovered by anyone who knows how to look.

Technical Summary: Maximal Subgroups of Free Projection- and Idempotent-Generated Semigroups with Applications to Partition Monoids

Problem Statement
The paper investigates the structure of maximal subgroups within two specific classes of free semigroups: the free idempotent-generated semigroup $IG(E)$ over a biordered set $E$ , and the free projection-generated regular $*$ -semigroup $PG(P)$ over a projection algebra $P$ . While the relationship between the idempotent structure of a semigroup and its maximal subgroups is well-established, determining the specific isomorphism types of these subgroups for concrete, complex semigroups remains a significant challenge.

The authors focus on the case where the underlying structures arise from the partition monoid $P_n$ . Specifically, they address the open problem of computing the maximal subgroups of $IG(E(P_n))$ and $PG(P(P_n))$ for projections and idempotents of rank $r \le n-2$ . A central motivation is to compare these free objects with the original semigroup $P_n$ and its submonoid of idempotent-generated elements, as well as to contrast the results with known cases like the full transformation monoid $T_n$ and the Temperley–Lieb monoid $TL_n$ .

Methodology
The paper employs a combination of combinatorial semigroup theory, group presentations, and topological rewriting techniques.

Reidemeister–Schreier Rewriting: The primary tool for deriving presentations of maximal subgroups is the Reidemeister–Schreier rewriting process, adapted for semigroups as developed in [81]. This method allows the authors to construct group presentations from semigroup presentations by analyzing the action of the semigroup on its $H$ -classes within a regular $D$ -class.
General Presentations: The authors first establish general presentations for the maximal subgroups of $PG(P)$ and $IG(E)$.
- For $PG(P)$, they derive presentations based on "linked diamonds" and "linked squares" of projections, reflecting the combinatorial topology of the projection algebra.
- For $IG(E)$, they utilize "singular squares" of idempotents.
- They demonstrate that $PG(P)$ is a natural homomorphic image of $IG(E)$, and their presentations explicitly reflect this quotient relationship.
Spanning Trees and Graph Theory: To simplify the presentations, the authors utilize spanning trees of the Graham–Houghton graph associated with the relevant $D$ -classes. By choosing specific spanning trees, they reduce the number of generators and relations, isolating the essential group structure.
Reduction to Transformation Monoids: A key strategic move is reducing the problem for the partition monoid $P_n$ to the known results for the full transformation monoid $T_n$ . The authors construct specific spanning trees and identify subsets of idempotents in $P_n$ that behave like those in $T_n$ , allowing them to "import" results regarding symmetric groups $S_r$ .
Twisted Partition Monoids: To explain the appearance of an infinite cyclic factor in the $IG(E)$ case, the authors connect their results to the twisted partition monoid $P^\Phi_n$ , which shares the same biordered set as $P_n$ but retains information about "floating components" in product graphs.

Key Contributions and Results

General Presentations: The paper provides explicit group presentations for the maximal subgroups of $PG(P)$ and $IG(E)$ in terms of generators and relations derived from linked diamonds, linked squares, and singular squares. These presentations reveal the topological nature of these groups as fundamental groups of complexes built from the underlying algebraic structures.
The Partition Monoid Case ( $P_n$ ): The authors compute the maximal subgroups for $P_n$ $P_{n}$ for ranks $0 \le r \le n-2$ $0 \leq r \leq n - 2$ :
- Theorem 1.1: The maximal subgroup of the free projection-generated semigroup $PG(P(P_n))$ corresponding to a projection of rank $r \le n-2$ is isomorphic to the symmetric group $S_r$ .
- Theorem 1.2: The maximal subgroup of the free idempotent-generated semigroup $IG(E(P_n))$ corresponding to an idempotent of rank $r \le n-2$ is isomorphic to the direct product $\mathbb{Z} \times S_r$ .
Rank 0 Case: For rank $r=0$ , the maximal subgroup of $PG(P(P_n))$ is trivial, while the maximal subgroup of $IG(E(P_n))$ is isomorphic to the infinite cyclic group $\mathbb{Z}$ .
Rank $n-1$ and $n$ : The paper notes that for $r=n-1$ , the subgroups are free groups of explicitly computable rank, and for $r=n$ , they are trivial.

Significance and Claims
The paper claims that these results resolve a long-standing open problem regarding the maximal subgroups of free idempotent-generated semigroups arising from diagram monoids.

Divergence from Transformation Monoids: Unlike the full transformation monoid $T_n$ , where the maximal subgroups of $IG(E(T_n))$ are isomorphic to $S_r$ (matching the subgroups of $T_n$ itself), the partition monoid $P_n$ exhibits a distinct behavior. The maximal subgroups of $IG(E(P_n))$ contain an extra infinite cyclic factor $\mathbb{Z}$ . The authors explain this via a connection to the twisted partition monoid $P^\Phi_n$ , suggesting that the "twisting" information (counting floating components) is intrinsically encoded in the biordered set of $P_n$ , even though the standard multiplication of $P_n$ forgets this data.
Structural Description: The determination of these maximal subgroups allows for a complete structural description of the regular $D$ $D$ -classes of $PG(P(P_n))$ $P G (P (P_{n}))$ and the regular homomorphic image $RIG(E(P_n))$ $R I G (E (P_{n}))$ .
- $PG(P(P_n))$ is described as "almost equal" to $P_n$ , differing only in the inflation of the maximal subgroups of the $D$ -class of rank $n-1$ (which become free groups) and the absence of non-identity bijections.
- $RIG(E(P_n))$ is described as "almost equal" to the twisted partition monoid $P^\Phi_n$ .
Algorithmic Properties: The authors deduce that both $PG(P(P_n))$ and $RIG(E(P_n))$ possess a solvable word problem, admit finite complete rewriting systems, and are residually finite.
Open Problems: The paper concludes by posing open questions, such as whether every group can arise as a maximal subgroup of a free projection-generated regular $*$ -semigroup, and whether the free idempotent-generated semigroup $IG(E(P_n))$ itself (which contains non-regular elements) has a solvable word problem.

The work establishes a rigorous framework for analyzing maximal subgroups in free objects generated by idempotents and projections, highlighting the subtle but profound differences between the free objects generated by diagram monoids versus transformation monoids.

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