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?