  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.
