Definition Semigroup

A semigroup is a pair $(S, \circ )$, consisting of a set $S$ and a mapping $\circ :S \times S \rightarrow S, (a,b) \mapsto a \circ b$ with the property of associativity:

\begin{align*}

a \circ (b \circ c) = (a \circ b) \circ c \quad \text{for all } a, b, c \in S

\end{align*}