Plactic monoids have their origin in Schensted's algorithm for computing the length of the longest increasing or decreasing subsequence of a given sequence. Knuth later pointed out the monoid structure of Schensted's purely combinatorial ideas. Recently, Plactic monoids and their associated monoid algebras have received a great deal of attention, but their combinatorial origin is sometimes overlooked.

In this talk, I will introduce Plactic monoids and survey some of their properties. I will then show how we can exploit the underlying combinatorial properties to obtain finite complete rewriting systems and biautomatic structures for Plactic monoids. In particular, this shows that Plactic algebras admit finite Gröbner--Shirshov bases.

This is joint work with Robert Gray and António Malheiro.