Identities and bases in the hypoplactic monoid

TL;DR

This paper investigates the algebraic identities of the hypoplactic monoid, demonstrating that all ranks greater than or equal to 2 satisfy the same identities and providing a finite axiomatic basis for the variety they generate.

Contribution

It establishes an embedding of higher-rank hypoplactic monoids into products of rank 2, and characterizes their identities with a finite basis.

Findings

All hypoplactic monoids of rank ≥ 2 satisfy the same identities.

The variety generated by the hypoplactic monoid has a finite basis.

A complete characterization of the identities is provided.

Abstract

This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it.