Definition 1.3.2.1. A monoid is a set $M$ equipped with a multiplication map
\[ m: M \times M \rightarrow M \quad \quad (x,y) \mapsto xy \]
which satisfies the following conditions:
- $(a)$
The multiplication $m$ is associative. That is, we have $x(yz) = (xy)z$ for each triple of elements $x,y,z \in M$.
- $(b)$
There exists an element $e \in M$ such that $ex=x=xe$ for each $x \in M$ (in this case, the element $e$ is uniquely determined; we refer to it as the unit element of $M$).