Diagrammatic Motzkin Monoids

• Diagrammatic Motzkin monoids are planar semigroups defined by noncrossing, partial pairings and isolated vertices, generalizing Motzkin paths.
• They feature a canonical factorization and a rigorous diagrammatic presentation using generators like Temperley–Lieb and planar rook elements.
• Their study unifies combinatorial enumeration, operadic structures, and cryptographic applications through detailed representation theory and ideal decomposition.

A diagrammatic Motzkin monoid is a semigroup of finite planar diagrams encoding noncrossing, partial pairings of vertices with possible isolated points, serving as a canonical example of a diagram monoid whose elements generalize the combinatorics of Motzkin paths. These monoids appear as submonoids of partition or Brauer monoids constrained by planarity and block size (≤2), and are central objects in the structure theory of planar diagram algebras, with deep connections to operad theory, representation theory, enumerative combinatorics, and cryptographic applications.

1. Structural Definition and Normal Forms

A diagrammatic Motzkin monoid $\mathcal{M}_n$ consists of all planar (noncrossing) set partitions of the points $\{1,\dots,n\}$ (top row) and $\{1',\dots,n'\}$ (bottom row) such that each block contains at most two elements. The possible blocks are:

• Transversals: $\{i, j'\}$ (a through strand)
• Upper hooks: $\{i, j\}$ (top-only arc)
• Lower hooks: $\{i',j'\}$ (bottom-only arc)
• Singletons: $\{i\}$ or $\{i'\}$ (isolated vertex)

Multiplication is diagrammatic: stack diagrams vertically, identify the bottom row of the upper with the top row of the lower, and connect blocks according to the matchings, deleting any resulting closed loops. This operation is associative and gives the Motzkin monoid its semigroup structure (Dolinka et al., 2015).

Every element admits a normal form

$$\alpha = \beta \cdot \lambda_{\operatorname{dom}(\alpha)} \cdot \rho_{\operatorname{codom}(\alpha)} \cdot \delta$$

where $\lambda_A, \rho_B$ are partial identities on subsets $A,B$, and $\beta, \delta$ are idempotent projections determined by the kernel and cokernel structure (Dolinka et al., 2015).

2. Diagrammatic Presentations and Generators

The Motzkin monoid can be faithfully represented in a diagrammatic fashion, with a rigorous presentation by generators and relations. Diagrammatically, it is generated by:

• Connectors $t_i$ (Temperley-Lieb type, creating a cup/cap at adjacent sites)
• Right and Left Planar Rook generators $r_i$, $l_i$ (removing specific vertices in the lower or upper row)
• Mixed elements $p_j$ (diagrams removing a specific vertex, or acting as partial identities)

The relations include:

• Temperley-Lieb relations: $t_i^2 = t_i$, $t_i t_j = t_j t_i$ for $|i-j|\ge 2$, $t_i t_j t_i = t_i$ for $|i-j|=1$
• Rook idempotents: $r_i^3=r_i^2$, $l_i^3=l_i^2$
• Braid-like and mixed relations, e.g. $t_i l_i = t_i r_i$, $r_i l_i = p_i$
• Commutation for disjoint indices and relations for “hop,” “slide,” “burrow,” “wallslide,” and “fuse-wire” moves that manipulate the positions of isolated vertices and edges (Posner et al., 2013).

Every diagram can be canonically factorized as $R \cdot T \cdot L$, where $R$ (right planar rook), $T$ (Temperley–Lieb), and $L$ (left planar rook) are from corresponding submonoids. This decomposition underlies the combinatorial structure and aids in algorithmic calculations within the monoid.

3. Operadic and Combinatorial Perspective

Diagrammatic Motzkin monoids admit a natural realization as operads arising from monoid-based constructions. Specifically, the suboperad of the operad $T(\mathbb{N})$ generated by $00$ and $010$ consists of those words $x = x_1 x_2 \ldots x_n$ satisfying $x_1=0$ and $|x_{i+1}-x_i|\le 1$ for all $i$. This correspondence encodes Motzkin paths: lattice paths from $(0,0)$ to $(n,0)$ with up, down, and horizontal steps (Giraudo, 2012, Giraudo, 2013).

Substitution in this operad mirrors diagram insertion:

$$(x_1,\dots, x_n)\circ_i (y_1,\dots, y_m) = (x_1,\ldots, x_{i-1}, x_i + y_1, \ldots, x_i + y_m, x_{i+1},\dots, x_n)$$

enforcing the noncrossing and unit-size block restrictions of Motzkin diagrams.

This operadic structure links Motzkin monoids with other classical objects (planar trees, Dyck paths, generalized Schröder paths) via different generator sets, providing a unifying combinatorial and algebraic framework (Giraudo, 2012, Giraudo, 2013).

4. Representation Theory and Cell Decomposition

Motzkin monoids are regular, $\mathcal{H}$-trivial semigroups whose ideals $I_r(\mathcal{M}_n)$ (elements of rank $\leq r$) are indexed by the number of through strands. Green's relations are determined by the domain, codomain, and rank (number of transversals); all maximal subgroups are trivial (Dolinka et al., 2015).

Cell modules in both the Motzkin monoid algebra and its associated diagram algebra are indexed by rank, with the cell basis given by all Motzkin diagrams of fixed rank. The simple module corresponding to rank $r$ has dimension $m(n,r)$, where $m(n,r)$ is Motzkin's trinomial recurrence: $$m(n, r) = m(n-1, r-1) + m(n-1, r) + m(n-1, r+1),\quad m(0,0)=1.$$ Minimal (idempotent) generating sets for ideals are completely characterized; $I_r(\mathcal{M}_n)$ is idempotent-generated if and only if $r < \lfloor n/2\rfloor$, and rank/idempotent rank equals

$$\mathrm{rank}(I_r(\mathcal{M}_n)) = m(n, r) + m'(n, r-1)$$

with $m'$ a related Riordan number sequence (Dolinka et al., 2015).

Motzkin algebras $\mathcal{M}_k(x)$ associated to the monoid admit a cellular structure in the sense of Graham–Lehrer, with left cell representations $\mathcal{C}_k^{(r)}$ arising from the algebra's canonical action on Motzkin paths of rank $r$. The determinants of the corresponding bilinear forms are rational expressions in Chebyshev polynomials, and the semisimplicity of $\mathcal{M}_k(x)$ is governed by the nonvanishing of the corresponding polynomials (Benkart et al., 2011).

5. Enumerative and Algorithmic Aspects

Enumeration of various Motzkin monoid features is algorithmically tractable via the combinatorial structure:

• The number of Motzkin diagrams of size $n$ is the $2n$th Motzkin number.
• Idempotents are classified using interface graphs, with a Motzkin diagram’s idempotence equivalent to all components of its interface graph being cycles, inactive paths, or active paths of even length. The number of idempotents lifting from a given minimal rank idempotent $\alpha$ is given by

$$|D^{-1}(\alpha)| = \prod_{\theta \in \Theta(\alpha)} (u_\theta(\alpha)\, l_\theta(\alpha) + 1)$$

where $u_\theta, l_\theta$ count outer hooks in each cycle component (Dolinka et al., 2015).

Efficient enumeration schemes for idempotents, elements of fixed rank, and certain step statistics on generalized Motzkin paths (including vertical steps) are developed; many are linked to Riordan arrays and classical sequences such as Catalan, Schröder, and Narayana numbers (Sun et al., 2022).

6. Maximal Subsemigroups and Congruence Lattice Structures

Diagrammatic Motzkin monoids have a finely stratified ideal and congruence structure:

• All maximal subsemigroups are classified via the removal of the unique unit or suitable Green’s classes from specific $J$-classes; the total number is $2^n+2n-3$ for $\mathcal{M}_n$ (East et al., 2017).
• The congruence lattice is not a chain: it consists of Rees congruences $R_q$ for ideals $I_q$ by rank, together with a “diamond” at the first two levels generated by additional congruences ($\lambda_0$, $\rho_0$, $\mu_0$) derived from retractions onto the minimal ideal $I_0$ from $I_1$ (East et al., 2017). This structure is explained via an ideal-retraction construction involving a liftable congruence on minimal ideals and is sensitive to the (trivial) maximal subgroup structure.

7. Representation Gap and Cryptographic Implications

Representations of Motzkin monoids exhibit a large representation gap, defined as the dimension of the smallest nontrivial linear representation (Arms, 8 Oct 2025). For the truncated Motzkin monoid $\mathcal{M}o_n^{\le \sqrt{n}}$, the gap satisfies

$$\mathrm{gap}_K(\mathcal{M}o_n^{\le \sqrt{n}}) \geq \Theta\left(n^{-3/2} 3^n\right)\,,$$

and

$$\lim_{n\to\infty} \left(\mathrm{gap}_K(\mathcal{M}o_n^{\le \sqrt{n}})\right)^{1/n} = 3.$$

The generic (semisimple) cell dimension is

$$\mathrm{ssdim}(L) = \sum_{t=0}^n \frac{k+1}{k+t+1}\binom{n}{k+2t}\binom{k+2t}{t}\,,$$

where $k$ is the number of through strands.

Thus, Motzkin monoids are resistant to linear decomposition attacks in cryptographic applications due to the absence of small-dimensional nontrivial representations (Arms, 8 Oct 2025). In contrast, non-pivotal (rigid) analogs, e.g., $r\mathsf{Mo}_n$, constructed by relaxing pivotal symmetry, yield much smaller minimal simple modules and lower gap ratios ($O(4^n)$ rather than $O(9^n)$), making them less suitable for cryptographic use (Stewart et al., 9 May 2025).

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The diagrammatic Motzkin monoid synthesis connects semigroup theory, diagram algebras, operad theory, and combinatorics, placing Motzkin monoids among the foundational objects for both mathematical structure and computational applications—especially where fine-grained enumerative, structural, and representation-theoretic properties are exploited.