Semigroup identities in the monoid of triangular tropical matrices. (Q2447232)

For each integer \(n\geq 2\), the author constructs a nontrivial semigroup identity that holds in the monoid of all \(n\times n\) upper triangular matrices over the tropical semiring \(\langle\mathbb R\cup\{-\infty\};\oplus,\otimes\rangle\), where \(a\oplus b:=\max\{a,b\}\), \(a\otimes b:=a+b\) for all \(a,b\in\mathbb R\cup\{-\infty\}\). The proof relies on a natural correspondence between tropical \(n\times n\) matrices and weighted digraphs with \(n\) vertices.