Tropical linear representations of the Chinese monoid (Q6047293)

The Chinese monoid which was explored by \textit{G. Duchamp} and \textit{D. Krob} [in: Words, languages and combinatorics II. Proceedings of the 2nd international conference, Kyoto, Japan, August 25-28, 1992. Singapore: World Scientific. 124--142 (1994; Zbl 0875.68720)] is a ternary monoid related to the plactic monoid and generated by the relations \(cba=cab=bca\) for totally ordered alphabets \(a\leq b \leq c\). Recall that a faithful linear representation of a semigroup \(S\) is a homomorphism \(\rho\) that takes any element in \(S\) to the multiplicative monoid of matrices of dimension \(n\times n\) over the tropical (max-plus) semiring \(\mathbb{T}\). If a homomorphism \(\rho\) is injective, then it is called faithful. In this paper, the authors give an inductive construction of a faithful linear representation for \(\mathrm{Ch}_{n+1}:= \langle a_1,\ldots,a_{n+1} \rangle\) out of a faithful representation for \(\mathrm{Ch}_{n}\) under which the number of generators equals \(n\) based on the concept of a faithful representation by matrcies of the plactic monoid \(\mathrm{Plc}_n\) over the tropical semiring. In fact, the appendices at the end of the paper provide a linear representation of the Chinese monoid of rank 2 and 3. The main result of the paper is as follows: The Chinese monoid \(\mathrm{Ch}_{n}\) of rank \(n\) has a faithful linear representation by tropical triangular matrices in (upper) triangular tropical matrices of size \( 2\cdot 3^{n-2}\), more precisely by block-diagonal triangular matrices with blocks of size 2. To prove this theorem, some lemmas are needed. Actually, in view of computational complexity, the main theorem also presents a beneficial tool for finding the canonical form of the Chinese monoid \(\mathrm{Ch}_{n}\).