  In algebraic terms, the insertion of $n$-powers in words may be modelled at
the language level by considering the pseudovariety of ordered monoids defined
by the inequality $1\le x^n$. We compare this pseudovariety with several other
natural pseudovarieties of ordered monoids and of monoids associated with the
Burnside pseudovariety of groups defined by the identity $x^n=1$. In
particular, we are interested in determining the pseudovariety of monoids that
it generates, which can be viewed as the problem of determining the Boolean
closure of the class of regular languages closed under $n$-power insertions. We
exhibit a simple upper bound and show that it satisfies all pseudoidentities
which are provable from $1\le x^n$ in which both sides are regular elements
with respect to the upper bound.
