Diagram Monoids: Algebraic & Combinatorial Insights

• Diagram monoids are finite semigroups whose elements are combinatorial diagrams representing set partitions, pairings, or connections on two parallel rows.
• They serve as foundational objects in algebraic combinatorics, with applications spanning representation theory, mathematical physics, and statistical mechanics through structured diagrammatic operations.
• Their rich structure, including Green’s relations, congruence lattices, and idempotent enumeration, enables precise presentations and algorithmic reductions.

A diagram monoid is a finite semigroup whose elements are combinatorial diagrams—typically, set partitions, pairings, or configurations of connections—on two parallel rows of points, with the monoid product defined by stacking diagrams, connecting vertices according to block structure, and reindexing or eliminating closed components formed in the middle row. Diagram monoids are foundational in algebraic combinatorics, with applications in commutative algebra, representation theory, and mathematical physics, due to their rich interaction with classical number sequences, their interplay with diagram algebras, and their structural features including Green’s relations, congruence lattices, and deep graphical connections.

1. Foundational Structure and Classes

Diagram monoids are defined on sets of diagrams representing equivalence relations or pairings on $2n$ vertices, arranged in a top (upper row) and bottom (lower row). The defining product "concatenates" two diagrams by stacking one atop another, connecting vertices from the upper row of the bottom diagram with the lower row of the top diagram, and removing loops entirely contained in the middle. Important classes include:

• Partition monoid $\mathcal{P}_n$: All set partitions of $\{1,\dots,n,1',\dots,n'\}$. Multiplication concatenates partitions, deleting middle-row-only components.
• Brauer monoid $B_n$: Submonoid of $\mathcal{P}_n$ where each block has exactly two elements (pairings).
• Partial Brauer monoid $PB_n$: All blocks of size at most two.
• Jones (Temperley–Lieb) monoid $J_n$: Planar submonoid of $B_n$ (no edge crossings).
• Motzkin monoid $M_n$: Planar submonoid of $PB_n$ with possible singleton vertices (isolated points).
• Planar partition monoid $PP_n$: Planar submonoid of all partitions (no crossings).
• Planar rook monoid $pRo_n$: Planar partial injective diagrams corresponding to non-crossing rook placements.

Block structure, planarity constraints, and allowed block sizes define each variety’s distinct combinatorics and representation-theoretic features (East et al., 2014, Dolinka et al., 2015, East et al., 2017).

2. Combinatorics and Enumeration

Diagram monoids encapsulate core combinatorial enumeration problems. The size and structural invariants of their ideals and classes are often governed by classical sequences:

Monoid	Number of elements	Idempotent Counts / Ranks
Partition	$|\mathcal{P}_n|=B_{2n}$	Ranks via Stirling and Bell numbers
Brauer	$|B_n|=(2n)!/(n! 2^n)$	Ranks via perfect matchings count $(2k-1)!!$
Jones	$|J_n|=C_n$	Ranks via Catalan and Fibonacci numbers
Motzkin	$|M_n|=M_n$	Ranks and idempotents via Motzkin/Riordan numbers

The ideals $I_r(M) = \{\alpha\in M:\text{rank}(\alpha)\leq r\}$ in these monoids are typically idempotent-generated, with minimal generating set sizes computed using explicit combinatorial sums:

• $\mathcal{P}_n$: $$\operatorname{rank}(I_r) = \operatorname{idrank}(I_r) = \sum_{j=r}^n S(n,j)\binom{j}{r}$$, $S(n,j)$—Stirling numbers of the second kind.
• $B_n$: $$\operatorname{rank}(I_r) = \operatorname{idrank}(I_r) = \binom{n}{2k}(2k-1)!!$$ for $r=n-2k$.

Minimal idempotent generating sets are characterized via balanced subgraphs of associated projection graphs, and in various cases correspond to matchings or cycle-factors in Johnson or path graphs (East et al., 2014, Dolinka et al., 2015).

3. Presentations, Decomposition, and Algorithms

Many diagram monoids admit presentations by generators and relations that encode their diagrammatic and planarity constraints. For example, the Motzkin monoid $M_n$ decomposes as $d = RTL$ where $R$ is in the right planar rook monoid, $T$ is a Temperley–Lieb diagram, and $L$ is in the left planar rook monoid, with explicit sets of generators (e.g., $t_i$, $r_i$, $l_i$) and intricate relations capturing commutativity, idempotence, and local rewrites. The full set of relations enforces planarity and the structure of empty vertices:

• $t_i^2=t_i$, $t_it_{i\pm1}t_i = t_i$
• $l_ir_i = r_{i+1}l_{i+1}$
• $r_il_j = l_j r_i$ when $|i-j| \geq 2$
• Various relations intertwining rook and Temperley–Lieb generators

Rewriting algorithms ("hop," "burrow," "slide," "wallslide," "fuse wire") reduce arbitrary words to normal form and are provably confluent and terminating, with correctness established via a counting argument that equates the number of standard words with the number of diagrams (Posner et al., 2013).

4. Green’s Relations, Congruences, and Lattice Structure

The fine structure of diagram monoids is captured with Green’s relations. In these semigroups:

• $\mathcal{R}$, $\mathcal{L}$: determined by domain/codomain, kernel/cokernel equality
• $\mathcal{D}$ (and $\mathcal{J}$ in the finite regular case): determined by rank

This structure allows for the description of maximal subsemigroups, congruence lattices, and ideal structure. The congruence lattices for finite diagram monoids are systematically described via a uniform construction involving ideals, retraction maps to minimal ideals, and the action of normal subgroups on maximal subgroups outside an ideal. All congruences are generated from:

• Rees congruences $R_I$
• Lifted congruences $\zeta_{I,\xi}$ for a retraction $I\to M$
• Relations $\nu_N$ from normal subgroups $N$ of maximal subgroups in a $J$-class Lattice diagrams (Hasse diagrams) exhibit diamond-shaped non-chain structures in the congruence order (East et al., 2017).

5. Enumeration and Classification of Idempotents

Idempotents are classified and enumerated in diagram monoids by translating diagrams to interface graphs—vertex- and edge-coloured multigraphs encoding domain, codomain, and block (hook) data. An idempotent corresponds precisely to a diagram whose interface graph decomposes into components of certain types (cycles, inactive paths, or active even-length paths). Enumeration leverages lifting from minimal-rank idempotents with fiber sizes given by cycle parameters.

For $M_n$, the number of idempotents is:

$$|E(M_n)| = \sum_{\widehat{\alpha}\in\Delta(M_n)} \prod_{\theta\in\Theta(\widehat{\alpha})} \left(u_\theta(\widehat{\alpha})\, l_\theta(\widehat{\alpha})+1\right)$$

where $u_\theta$, $l_\theta$ count upper/lower outer hooks in each cycle component $\theta$ (Dolinka et al., 2015). Algorithmic enumeration exploits this local structure and outperforms naïve or regular $*$-semigroup enumeration methods.

6. Representation Theory and Cellular Structure

Diagram monoids underpin a family of diagram algebras, which—under suitable scalars and twistings—become cellular algebras in the sense of Graham and Lehrer. Their representation theory is controlled by cell modules indexed by $J$-classes and irreducible representations of maximal subgroups.

Dimensional formulas express the sizes of cell modules and irreducible modules in terms of combinatorial data:

• Partition algebra: $$\dim(A_n^\mu) = (\sum_{j=|\mu|}^n S(n,j)\binom{j}{|\mu|}) \cdot (|\mu|! / \prod_{b\in\mu} h(b))$$
• Brauer algebra: $$\dim(B_n^\mu) = (n!/(2^k k! r!)) \cdot (|\mu|!/ \prod_{b\in\mu} h(b))$$, $r=n-2k$
• Temperley–Lieb algebra: $\dim(J_n^r) = \frac{r+1}{n+1} \binom{n+1}{k}$, $r=n-2k$
• Motzkin algebra: analogues with structure constants given by Motzkin numbers and hooks (East et al., 2014, Dolinka et al., 2015).

Connection to planar algebras, quantum invariants, and number theory (Bell, Catalan, Fibonacci, Motzkin numbers) is pervasive.

7. Applications and Generalizations

Diagram monoids arise universally in algebraic and combinatorial representation theory, statistical mechanics (Temperley–Lieb, Brauer, partition algebras), random processes, quantum algebra, and categorical algebra (e.g., string diagram rewriting, tape diagrams). They admit generalizations:

• Boolean reflection monoids (related to reflection groups and quiver mutations) (Duan et al., 2017)
• Okada monoid/algebra (via labeled Temperley–Lieb diagrams and Fibonacci combinatorics) (Hivert et al., 25 Apr 2024)
• Rigid (non-pivotal) diagram monoids with cryptographic implications (Stewart et al., 9 May 2025)
• Transformation representations with explicit minimum degree formulas, e.g., for $\mathcal{P}_n$: $$\deg(\mathcal{P}_n) = 1 + \frac{B(n+2)-B(n+1)+B(n)}{2}$$ (Cirpons et al., 22 Nov 2024)

They provide deep connections between algebraic structure, graph theoretical representations (Cayley, Graham–Houghton graphs), and computational complexity.

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Diagram monoids thus constitute a central combinatorial-algebraic object class, characterized by diagrammatic operations, rich enumerative and algebraic structure, and far-reaching interactions with various branches of mathematics and mathematical physics. Their paper marries graphical representations, semigroup theory, and deep combinatorics, providing a flexible foundation for the analysis of both classical and modern algebraic systems.