When a cube equals a square

Question

An element $x$ of a monoid is called idempotent if $x^2 = x$. Is there a word for an element $x$ such that $x^3=x^2$? (Of course it follows that $x^2$ is idempotent.)

Answer 1

Suppose there exist integers such that $x^i = x^{i+p}$. The minimal $i$ and $p$ with this property are called respectively the index and the period of $x$. If $p = 1$, $x$ is said to be aperiodic. Thus an element satifying $x^2 = x^3$ is an aperiodic element of index $2$.