Identities and bases in the sylvester and Baxter monoids

TL;DR

This paper investigates the identities of sylvester and Baxter monoids, showing they satisfy the same identities across ranks and providing finite bases for their varieties, advancing understanding of their algebraic structure.

Contribution

It introduces embeddings of higher-rank monoids into rank-2 products and characterizes the identities and finite bases of the varieties they generate.

Findings

All monoids of the same family with rank ≥ 2 satisfy identical identities.

The varieties generated by these monoids have finite axiomatic rank.

A finite basis for the identities of these monoids is provided.

Abstract

This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, 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 varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.