The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep connections to several areas of mathematics, in particular, to the theory of symmetric functions. An actively-studied problem is the identities satisfied by the plactic monoids of finite rank. It is known that there is no “global" identity satisfied by the plactic monoid of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic monoid hypo (the monoid of quasi-ribbon tableaux, connected with quasisymmetric functions), sylvester monoid sylv (the monoid of binary search trees) and Baxter monoid (pairs of twin binary search trees, connected with Baxter), satisfy global identities, and the shortest identities have been characterized.

This talk will present new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank greater than 3 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This shows that all these monoids satisfy the same identities. We then give a complete characterization of the identities satisfied by hypo. Then, we show that the variety generated by hypo has finite axiomatic rank, and present a finite basis for the equational theory of hypo. We also show how to embed the hypoplactic monoids into a subdirect product of subdirectly irreducible monoids.