Is there a notion of "sequential" idempotence? TL;DR: I have a definition, and I'm wondering if it already has a name or has been studied.

Suppose we have a sequence of operations (or if we want to be mathematical, functions whose domains and ranges are all the same) $f_1, f_2, ..., f_n$.

I'll call the sequence idempotent if the composition $f_1f_2...f_n$ is idempotent.

But I call the sequence strongly idempotent if any initial subset of the operations $f_1...f_k$ is idempotent for all $1 \leq k \leq n$.

For example, if we take the domain of integers, set $f_1(x) = x + 1$ and $f_2(x) = 0$, then the sequence $(f_1, f_2)$ is idempotent but not strongly idempotent.

My question is, does this definition/idea already have a name? Are there any known useful properties of strong idempotency, or is it known to be useful in any particular areas?
Your question would probably fit better on MathsStackExchange, but here is an answer.

I don't think there is any specific name for your definition. Since any semigroup is isomorphic to a transformation semigroup, your question is about finitely generated semigroups that are quotients of the semigroup $S_n$ with presentation
$$
\langle a_1, \ldots, a_n \mid a_1^2 = a_1, (a_1a_2)^2 = a_1a_2, \ldots, (a_1 \dotsm a_n)^2 = a_1 \dotsm a_n \rangle
$$
It turns out that $S_n$ is infinite for $n &gt; 1$, but I don't know whether it has been studied in the literature.