Grammic Monoid: Combinatorial and Tropical Perspectives

• Grammic monoid is an infinite monoid built on a variant of the row‐insertion algorithm in Young tableau theory that discards bumped letters.
• It is finitely presented for three generators with classical Knuth relations and a unique extra relation (3212 = 2132), ensuring a confluent rewriting system.
• It embeds faithfully into upper‐triangular tropical matrices, linking combinatorial tableau theory with tropical semigroup identities.

A grammic monoid is an infinite monoid constructed from the combinatorics of Young tableaux, specifically encoding the behavior of the row-insertion algorithm when each bumped letter is discarded rather than recursively inserted. The fundamental object is the grammic congruence, defined on the free monoid over a finite ordered alphabet via the partial action of row-insertion on all weakly increasing words ("rows"). For the case of three generators, the grammic monoid admits an explicit finite presentation as a quotient of the free monoid by the classical Knuth relations and a single additional relation. More generally, the congruence is characterized by the statistics of weakly increasing subsequences within every alphabet interval, and the grammic monoid of rank $n$ is faithfully embedded as a submonoid of the $n \times n$ upper-triangular tropical matrices. This fusion of combinatorial, algebraic, and semigroup-theoretic approaches situates the grammic monoid at the interface of tableau theory and tropical semigroup identities.

1. Construction via Row-Insertion and Grammic Congruence

Let $A = \{1 < 2 < 3\}$ be a totally ordered alphabet. A row is any finite non-decreasing word in $A^*$ (e.g., $11233$). The action underlying the grammic monoid is Schensted's row-insertion: given a row $R = b_1 \cdots b_p$ and a letter $b \in A$, $b$ is inserted by:

• If $b > b_p$, set $R \cdot b = b_1 \cdots b_p b$.
• Otherwise, find the leftmost $b_i > b$, replace $b_i$ by $b$, and (classically) recursively insert the bumped $b_i$ into the next row up. In the grammic action, the bumped letter is discarded.

The action of the free monoid $A^*$ on the set of all such rows by iterated row-insertion (with bump discards) defines a congruence $\equiv_{\mathrm{gram}}$ on $A^*$:

$$u \equiv_{\mathrm{gram}} v \iff \text{for all rows } R,~R \cdot u = R \cdot v$$

The grammic monoid $M$ is the quotient $A^*/\equiv_{\mathrm{gram}}$ (Choffrut, 2022, Johnson et al., 27 Nov 2025).

2. Presentations and Relations: The Three-Generator Case

For $A = \{1,2,3\}$, the grammic monoid $M$ has a presentation by generators and relations reflecting both classical tableau combinatorics and an additional structure:

• Generators: $1, 2, 3$.
• Defining Relations:
  • Knuth relations (as in the plactic monoid):
  • $\text{K1: } bac = bca$ for $a<b \leq c$,
  • $\text{K2: } acb = cab$ for $a \leq b < c$.
  • Unique extra relation arising for three generators:
  • $\text{E: } 3212 = 2132$.
  • Thus,

$$M = \langle 1,2,3 \mid \text{K1},\text{K2},\,3212 = 2132\rangle$$

Every grammic congruence class reduces to a unique row-normal form via these rules—read off as the concatenation of the rows of the Schensted tableau, from top to bottom, with each row being non-decreasing. The rewriting system is confluent and terminating (Choffrut, 2022).

3. Combinatorial and Algebraic Characterizations

A general and robust characterization of the grammic congruence is given via statistics of weakly increasing subsequences:

• For any $w \in [n]^+$ and $1 \leq p \leq q \leq n$, let $w_{p,q}$ be the maximal length of a weakly increasing subsequence of $w$ consisting only of letters in $\{p,\ldots,q\}$.
• Theorem: $u \equiv_{\mathrm{gram}} v$ if and only if $u_{p,q} = v_{p,q}$ for all $1 \leq p \leq q \leq n$.

The tropical representation $\varphi_n$ sends each word $w$ to an $n \times n$ upper-triangular matrix $(\varphi_n(w))_{p,q} = w_{p,q}$. The kernel of this representation is precisely $\equiv_{\mathrm{gram}}$, so $\G_n \cong \im(\varphi_n) \subseteq \mathrm{UT}_n(\mathbb{T})$ (the monoid of $n \times n$ upper-triangular matrices over the tropical semiring) (Johnson et al., 27 Nov 2025).

4. Structural Properties, Identities, and Equational Theory

The grammic monoid $M$ is infinite for $|A| \geq 3$, with word problem decidable via its finite convergent presentation (Choffrut, 2022). The growth of the Cayley graph is polynomially bounded but not ultimately periodic. The grammic congruence is strictly coarser than the plactic congruence: $3212 \equiv_{\mathrm{gram}} 2132$, yet $3212 \neq_{\mathrm{plactic}} 2132$.

Semigroup identities for $\G_n$ agree with those of $\mathrm{UT}_n(\mathbb{T})$. In particular, for $n=3$, Volkov demonstrated that the grammic and plactic monoids with three generators satisfy exactly the same identities:

$M \models w = w' \iff N_2 \models w = w'$

where $N_2$ is the grammic quotient of the plactic monoid (Volkov, 2022). For each finite $n$, the shortest non-trivial identity has length $n$; for infinite rank, no non-trivial identity survives (Johnson et al., 27 Nov 2025).

5. Connections to Tableau Theory and Other Combinatorial Monoids

The grammic monoid arises from a variant of the row-insertion process in Young tableau theory, with each bump discarded rather than recursively inserted. Classical plactic and grammic monoids thus both encode tableaux combinatorics but differ in how insertion cascades are handled. Unlike the column-insertion (stylic) monoid, which is finite, the grammic monoid is infinite for $|A|\ge3$ due to unbounded row growth, but it retains much of the structural flavor of tableau insertion monoids (Choffrut, 2022).

Notably, the grammic monoid with three generators is a quotient of the plactic monoid by a single extra relation ($bacb = cbab$), and yet their identities coincide, signifying a surprising rigidity in the equational theory imposed by Young tableau combinatorics (Volkov, 2022).

6. Further Characterizations and Open Problems

Compatibility properties of the grammic congruence mirror those for the plactic congruence:

• Restriction to subintervals of the alphabet,
• Compatibility with packing and standardization,
• Invariance under Schützenberger involution.

Whether a finite presentation exists in the general rank-$n$ case is open, as is the finite basis problem for the associated varieties. The correspondence between the grammic monoid and its tropical matrix representation yields a unifying algebraic perspective, situating $\G_n$ in the same variety as $\mathrm{UT}_n(\mathbb{T})$ (Johnson et al., 27 Nov 2025).

Current research focuses on extending the equational equivalence of grammic and plactic monoids to higher rank, explicit determination of minimal non-trivial identities, and broader conceptual frameworks connecting tableau-based monoids with other insertion-like structures (e.g., hypoplactic, Baxter monoids) (Volkov, 2022).