<p>Your question would probably fit better on MathsStackExchange, but here is an answer.</p>

<p>I don't think there is any specific name for your definition. Since any semigroup is isomorphic to a <a href="https://en.wikipedia.org/wiki/Transformation_semigroup" rel="nofollow noreferrer">transformation semigroup</a>, your question is about finitely generated semigroups that are quotients of the semigroup <span class="math-container">$S_n$</span> with <a href="https://en.wikipedia.org/wiki/Presentation_of_a_monoid" rel="nofollow noreferrer">presentation</a>
<span class="math-container">$$
\langle a_1, \ldots, a_n \mid a_1^2 = a_1, (a_1a_2)^2 = a_1a_2, \ldots, (a_1 \dotsm a_n)^2 = a_1 \dotsm a_n \rangle
$$</span>
It turns out that <span class="math-container">$S_n$</span> is infinite for <span class="math-container">$n &gt; 1$</span>, but I don't know whether it has been studied in the literature.</p>

CC BY-SA 4.0
2019-11-21T08:38:22.513Z
2019-11-21T08:38:22.513Z
J.-E. Pin
<p>TL;DR: I have a definition, and I'm wondering if it already has a name or has been studied.</p>

<p>Suppose we have a sequence of operations (or if we want to be mathematical, functions whose domains and ranges are all the same) <span class="math-container">$f_1, f_2, ..., f_n$</span>.</p>

<p>I'll call the sequence <em>idempotent</em> if the composition <span class="math-container">$f_1f_2...f_n$</span> is idempotent.</p>

<p>But I call the sequence <em>strongly idempotent</em> if any initial subset of the operations <span class="math-container">$f_1...f_k$</span> is idempotent for all <span class="math-container">$1 \leq k \leq n$</span>.</p>

<p>For example, if we take the domain of integers, set <span class="math-container">$f_1(x) = x + 1$</span> and <span class="math-container">$f_2(x) = 0$</span>, then the sequence <span class="math-container">$(f_1, f_2)$</span> is idempotent but not strongly idempotent.</p>

<p>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?</p>

CC BY-SA 4.0
2019-11-20T22:17:10.213Z
If you want your property to hold whatever is the product order, then it means that the semigrouo generated by your functions is regular.
C.P.
CC BY-SA 4.0
2019-11-21T08:10:54.007Z
@C.P. A regular semigroup is not necessarily idempotent.
J.-E. Pin
CC BY-SA 4.0
2019-11-21T08:44:00.857Z
A finite aperiodic regular semigroup is, though. I don't know for infinite. Do you?  I was imprecise indeed. Thank you for the: "pan sur le bec" :).
C.P.
<terminology>
Is there a notion of "sequential" idempotence?
Dan Staley
# Is there a notion of "sequential" idempotence?

(Asked by: username_0 on 2019-11-20)

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?

**Tags:** terminology

> **Comments:**
> **username_1**: If you want your property to hold whatever is the product order, then it means that the semigrouo generated by your functions is regular.  
> **username_2**: @username_1 A regular semigroup is not necessarily idempotent.  
> **username_1**: A finite aperiodic regular semigroup is, though. I don't know for infinite. Do you?  I was imprecise indeed. Thank you for the: "pan sur le bec" :).  

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## 1 answers

### Answer by username_2 on 2019-11-21. Score: 4

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](https://en.wikipedia.org/wiki/Transformation_semigroup), your question is about finitely generated semigroups that are quotients of the semigroup $S\_n$ with [presentation](https://en.wikipedia.org/wiki/Presentation_of_a_monoid)
$$
\langle a\_1, \ldots, a\_n \mid a\_1^2 = a\_1, (a\_1a\_2)^2 = a\_1a\_2, \ldots, (a\_1 \dotsm a\_n)^2 = a\_1 \dotsm a\_n \rangle
$$
It turns out that $S\_n$ is infinite for $n > 1$, but I don't know whether it has been studied in the literature.


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