Is there a notion of "sequential" idempotence?

Question

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?

Answer 1

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 $n > 1$, but I don't know whether it has been studied in the literature.