Equational theories of upper triangular tropical matrix semigroups
Tropical mathematics now interact with many fields and applications, from algebraic geometry to optimization, phylogenetics to economics. The very origin of the word ``tropical'' came from semirings and automata theory. A nice historical overview is given by \textit{J.-E. Pean} [Tropical semirings, idempotency. Bristol. 50--69 (1994)]. This paper falls into this line of work. A simplistic way to view this line of work is to take the tropical semiring as an interesting algebra, and revisit classical questions for classical objects in semigroup theory over this algebra. For concreteness, one could take the min-plus semiring, where minimum replaces addition and addition replaces multiplication. The set of \(n \times n\) matrices equipped with matrix multiplication over the min-plus semiring becomes a semigroup. Now one could ask, for example, does the semigroup of \(n \times n\) tropical matrices have a finite basis? Does this semigroup have interesting identities? One can also consider other semigroups, such as those of upper triangular \(n \times n\) tropical matrices, denoted \(UT_n\). It turned out that \(UT_n\) is particularly interesting, as their identities have strong relations with the bicyclic and plactic monoid, which are other classical obects in semigroup theory. In particular, various researchers have resolved open questions related to the bicyclic and plactic monoid by first translating the questions in terms of \(UT_n\), and then use combinatorial or geometric techniques to answer them. See the citations on page 3, second paragraph, of this paper. This paper builds upon previous work and resolved the finite basis problem for the semigroups of upper triangular matrices \(UT_n\) for \(n = 2,3\). It also stated interesting open problems for \(n \geq 4\).
scientific article; zbMATH DE number 7382708
scientific article