Is there a term for the closure of all endomorphism under composition?

Question

Basically if I have a set of endomorphisms $M$ and I define an operator $ \bigcirc M = \{ m_1 \circ m_2 : m_1,m_2 \in M \} $. I can define the closure of $ \bigcirc M $ as the least fixed point of $ N \mapsto M \cup N \cup \bigcirc N $.

Is there a standard term for this closure?

Answer 1

One standard term for this would be "the semigroup generated by $M$". It is the smallest sub-semigroup of the full set of endomorphisms which contains $M$.

Answer 2

Another term could be the transitive closure of $M$.