Power Quotients of Plactic-like Monoids

In this paper we describe the quotients of several plactic-like monoids by the least congruences containing the relations $a^{\sigma (a)}=a$ with $\sigma (a)\ge 2$ for every generator $a$. The starting point for this description is the recent paper of Abram and Reutenauer about the so-called stylic monoid which happens to be the quotient of the plactic monoid by the relations $a^2=a$ for every letter $a$. The plactic-like monoids considered are the plactic monoid itself, the Chinese monoid, and the sylvester monoid. In each case we describe: a set of normal forms, and the idempotents; and obtain formulae for their size.