Let $M$ be a monoid, and let $x\in M$. One says that $x$ is *periodic* if
$$x^{i+j}=x^j$$
for some integers $i\geq 1$ and $j\geq 0$.


An easy division algorithm argument shows that if $m$ is the smallest value of $i$ where this happens (for some $j$), and similarly $n$ is the smallest value of $j$ where this happens (for some $i$), then $x^{m+n}=x^n$. (So those minimal values work together.)


Moreover, given such an $m$ and $n$, the displayed equality holds if and only if $m|i$ and $n\leq j$.


Question 1: Is there a standard reference for these basic facts in the monoid setting?


Question 2: Is there a standard name for $m$ and $n$?


Question 3: Are there special names for the periodic property when $n=0$ or when $n=1$? (I've seen them called "torsion units" and "potents" [generalizing "idempotents"] in the ring-theoretic setting.)



More vocabulary for periodic elements in monoids
Question 1: See Clifford and Preston, volume 1.


Question 2: $m$ is the period, $n$ is called the index of the element. See this [Wikipedia text](https://en.wikipedia.org/wiki/Monogenic_semigroup).


Question 3: If $n=1$, the element is called a group element of finite order. If $n=0$, it is called a unit of finite order. Or you can just call it an element of index 1 (resp. 0).


