I'm reading A Short Course on Operator Semigroups by Engel and Nagel, and they say on page 2:
A family $\left(T(t)\right)_{t \geq 0}$ of bounded linear operators on a Banach space is called a strongly continuous (one parameter) semigroup if it satisfies the functional equation \begin{align} T(t + s) &= T(t)T(s) \text{ for all} t,s \geq 0, \\ T(0) &= I \end{align} and is strongly continuous... [in a way they go on to define].
In other words, this `semigroup' has an identity. But I thought a semigroup with an identity is called a monoid? So I'm wondering if I'm wrong, or if this field is just called operator semigroup theory for historical reasons, even though a modern algebraist would call these objects monoids?