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 $a^{\sigma (a)}=a$ for $\sigma (a)\ge 2$ for every generator $a$. The starting point of this paper is the recent paper of Reutenauer and the first author about the stylic monoid. This monoid is the quotient of the plactic monoid by the relations $a^2=a$ for every generator $a$. The plactic-like monoids considered here are the plactic monoid itself, the Chinese monoid, the hypoplactic monoid, the sylvester monoid, the metasylvester and hyposylvester monoids. In each case we describe a set of normal forms and the idempotents. We obtain formulae for their cardinalities and we describe their isomorphism types.