Finite basis problems for stalactic, taiga, sylvester and Baxter monoids

TL;DR

This paper investigates the finite basis property of certain monoids derived from tableau combinatorics, establishing conditions under which these monoids are finitely based and identifying their identities.

Contribution

It provides three sufficient conditions for semigroups to be finitely based and applies them to classify stalactic, taiga, sylvester, and Baxter monoids of rank ≥ 2.

Findings

All stalactic and taiga monoids of rank ≥ 2 are finitely based and share the same identities.

All sylvester monoids of rank ≥ 2 are finitely based and share the same identities.

All Baxter monoids of rank ≥ 2 are finitely based and share the same identities.

Abstract

Stalactic, taiga, sylvester and Baxter monoids arise from the combinatorics of tableaux by identifying words over a fixed ordered alphabet whenever they produce the same tableau via some insertion algorithm. In this paper, three sufficient conditions under which semigroups are finitely based are given. By applying these sufficient conditions, it is shown that all stalactic and taiga monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities and that all Baxter monoids of rank greater than or equal to $2$ are finitely based and satisfy the same identities.