Monoid

A monoid $(M,*)$ is a unital magma such that the binary operation $*$ is associative, which means that for all $x,y,z\in M$, one has the equality:
$$(x*y)*z=x*(y*z)$$Examples:

$\bullet$ The magma $(\Z,-)$ is not a monoid since $(1-2)-3=-4\neq 2=1-(2-3)$.

$\bullet$ The magma $(\Z,+)$ is a monoid since $(a+b)+c=a+(b+c)$ for all $a,b,c\in\Z$ and $0$ is an identity element.