Upper Triangular Tropical Matrices

• Upper triangular tropical matrices are n×n matrices over the tropical semiring with all entries below the diagonal set to –∞, playing a key role in combinatorics and tropical geometry.
• They satisfy distinct semigroup identities, including generalized Adjan-type identities, which reveal non-finite basis properties and an increasing chain of varieties.
• Their study leverages tropical polynomial criteria and Newton polytope techniques to enable efficient algorithmic verification of identity equivalence.

An upper triangular tropical matrix is an $n\times n$ matrix over the tropical semiring $$\mathbb T = (\mathbb{R} \cup \{-\infty\}, \oplus, \otimes)$$, where the semiring operations are $a \oplus b = \max\{a,b\}$ and $a \otimes b = a + b$, and all entries below the main diagonal are $-\infty$. The structure and identities of the monoid $UT_n(\mathbb{T})$ of such matrices under tropical multiplication are the focus of extensive research due to their connections with combinatorics, semigroup theory, and tropical geometry.

1. Definition and Basic Properties

Given the tropical semiring $\mathbb{T}$, the monoid $UT_n(\mathbb{T})$ is the set of $n\times n$ upper triangular matrices with entries in $\mathbb T$, i.e., matrices $A = (A_{ij})$ such that $A_{ij} = -\infty$ whenever $i > j$. Matrix addition and multiplication are:

• $(A \oplus B)_{ij} = \max\{A_{ij}, B_{ij}\}$
• $$(A \otimes B)_{ij} = \max_{1 \leq k \leq n}(A_{ik} + B_{kj})$$

The multiplicative identity is the diagonal matrix with $0$ on the diagonal and $-\infty$ elsewhere. $UT_n(\mathbb{T})$ is closed under tropical matrix addition and multiplication, forming a monoid (Johnson et al., 2019).

2. Semigroup Identities and Non-finite Basis Results

A distinctive aspect of $UT_n(\mathbb{T})$ is its semigroup identities. For $n=2$, $UT_2(\mathbb{T})$ satisfies all identities of the bicyclic monoid; in fact, the bicyclic monoid $\mathcal{B}$ canonically embeds in $UT_2(\mathbb{T})$ (Chen et al., 2015, Daviaud et al., 2016). The Adjan identity, a cornerstone in the theory of semigroup varieties, is a key identity for $UT_2(\mathbb{T})$:

$$(xy)(yx)(xy)(xy)(yx) \ \approx\ (xy)(yx)(yx)(xy)(yx)$$

Generalizing, $UT_2(\mathbb{T})$ satisfies for each $n\geq 1$ the identity:

$u_n = v_n$

where $u_n$ and $v_n$ are obtained from Adjan's identity via variable substitutions. Any monoid containing a bicyclic submonoid and satisfying all such $u_n = v_n$ is non-finitely based; thus, $UT_2(\mathbb{T})$ admits no finite identity basis and has infinite axiomatic rank (Chen et al., 2015).

For $n>2$, research establishes the existence of nontrivial semigroup identities for $UT_n(\mathbb{T})$. For example, for each $n$, there exists an identity that distinguishes $UT_n$ from $UT_{n+1}$, implying a strictly increasing chain of varieties:

$$\mathrm{Var}(UT_1(\mathbb{T})) \subsetneq \mathrm{Var}(UT_2(\mathbb{T})) \subsetneq \cdots$$

The explicit construction of such identities exploits combinatorial tools (see Section 4) (Aird, 2021, Izhakian, 2013).

3. Tropical Polynomial and Polyhedral Criteria for Identities

A foundational result is the characterization of identities in $UT_n(\mathbb{T})$ via tropical polynomials. For $n=2$, the identity $w = v$ holds in $UT_2(\mathbb{T})$ if and only if for every letter $s$ of the alphabet, the associated tropical polynomials $f_s^w$ and $f_s^v$ are equivalent as functions, where:

$$f_s^w(x_t\ |\ t\in\Sigma) = \bigoplus_{w_i=s} \ \bigotimes_{t\in\Sigma} x_t^{\lambda_t^w(i-1)}$$

Here, $\lambda_t^w(k)$ counts occurrences of $t$ in the first $k$ letters of $w$ (Daviaud et al., 2016). For $n>2$, one examines more elaborate multivariate tropical polynomials encoding the combinatorics of all subwords of length up to $n-1$. The identity $w = v$ holds in $UT_n(\mathbb{T})$ if and only if for each such subword and each path in the $n$-chain, the corresponding tropical polynomials for $w$ and $v$ are equivalent.

A geometric viewpoint replaces polynomial equivalence with Newton polytope equality: $w \sim_n v$ in $UT_n$ if and only if, for all subwords $u$, the Newton polytopes of the corresponding tropical polynomials for $w$ and $v$ coincide (Johnson et al., 2018).

4. Construction and Quantitative Aspects of Semigroup Identities

Identity construction in $UT_n(\mathbb{T})$ is tied to combinatorics. For each $n\geq 2$, one constructs a word $w$ such that every subword of length $n-1$ appears as a factor and such that certain powers do not occur as factors. Then $UT_n(\mathbb{T})$ satisfies:

$w\, a\, w[ab,ba] = w\, b\, w[ab,ba]$

where $w[a,b]$ denotes the evaluation of $w$ by substituting $a$ and $b$ as indicated (Aird, 2021). For example, when $n=3$ and $w=ab^2a^2b$, this yields a concrete identity separating $UT_3$ from $UT_4$.

Taylor (Izhakian, 2013) establishes bounds on the minimal length of 2-variable identities in $UT_n(\mathbb{T})$ via the formula for Fibonacci numbers, reflecting the number of $(n-1)$-power words. Furthermore, recursive identities based on Adjan's construction and combinatorial encodings via Young tableaux provide efficient presentations for identities in $UT_n$ (Cain et al., 2017).

5. Connections with Other Monoids: Embeddings and Variety Structure

$UT_n(\mathbb{T})$ plays a central role in representing and distinguishing varieties of semigroups related to combinatorics and representation theory. Critical connections are:

• Bicyclic Monoid: $UT_2(\mathbb{T})$ and the bicyclic monoid share exactly the same identities. The embedding of the bicyclic monoid in $UT_2$ is explicit and preserves identity structure (Chen et al., 2015, Daviaud et al., 2016, Nyberg-Brodda, 2022).
• Plactic Monoids: For each $n\geq1$, the rank-$n$ plactic monoid $P_n$, defined via the Knuth relations on tableaux, embeds faithfully in $UT_{2^n}(\mathbb{T})$. The variety generated by $UT_3(\mathbb{T})$ coincides precisely with that generated by the plactic monoid $P_3$ (Johnson et al., 2019, Cain et al., 2017). These embeddings are constructed using actions on configuration tableaux and the factorization properties of tropical matrix semigroups.
• Chain-structured Semigroups: $UT_n(\mathbb{T})$ and all chain-structured tropical matrix semigroups of length $n$ share identical identities, a consequence of Birkhoff’s HSP theorem and the polynomial criteria (Daviaud et al., 2016).

The following table summarizes some key embedding and variety results:

Structure	Embedding Target	Variety Identity Profile
Bicyclic monoid ($\mathcal{B}$)	$UT_2(\mathbb{T})$	Same as $UT_2(\mathbb{T})$
Plactic monoid $P_n$	$UT_{2^n}(\mathbb{T})$	All $P_n$ identities in $UT_n$
Chain-structured semigroup	$UT_n(\mathbb{T})$	Identical for fixed $n$

6. Algorithmic and Geometric Aspects

A principal achievement in the theory is the algorithmic verification of semigroup identities via polyhedral and tropical polynomial methods. For fixed $n$, it is possible to decide in polynomial time (in the length of the identity and alphabet size) whether a given word identity holds in $UT_n(\mathbb{T})$, by reducing the problem to testing equivalence of tropical polynomials via linear programming (Daviaud et al., 2016, Johnson et al., 2018). This geometric realization leverages Newton polytopes, distributive lattice structures in the binary case, and signature algorithms capable of efficiently exploring the space of possible identities.

In $UT_2(\mathbb{T})$, classes of equivalence under semigroup identities correspond to intervals in the lattice of lattice paths, and the minimal lengths of nontrivial identities can be established precisely (e.g., length 10 for minimal Adjan-type identities) (Johnson et al., 2018).

7. Classification, Applications, and Open Questions

Upper triangular tropical matrix monoids admit a precise classification regarding which one-relation monoids they can represent. With a finite list of exceptions, all embeddable one-relation monoids are described explicitly, emphasizing the tight constraints imposed by the tropical structure and its combinatorial consequences (Nyberg-Brodda, 2022).

Applications of upper triangular tropical matrices include the representation theory of monoids, the combinatorics of Young tableaux and plactic algebras, and investigations of semigroup varieties arising from idempotent and max-plus algebraic structures.

Open questions include:

• Whether $UT_n(\mathbb{T})$ is non-finitely based for $n>2$ (affirmative for $n=2$) (Chen et al., 2015).
• The determination of whether the variety generated by the plactic monoid of rank 4 coincides with that generated by $UT_4(\mathbb{T})$.
• The structure and finiteness properties of submonoids defined by graph or weight constraints (Aird, 2021).

The study of upper triangular tropical matrices thus forms a hub linking tropical algebra, semigroup identities, combinatorial representation theory, and computational algebra, with ongoing developments at several theoretical frontiers.