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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 ...
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