Associative and Commutative Binary Operations, and Neutral Elements

In the setting of Sets and Maps: Ordinary Notation, let $\ast$ be a binary operation on a set $X$.

¶ $\ast$ is associative if $(x\ast y)\ast z=x\ast(y\ast z)$ for all $x,y,z\in X$.

¶ $\ast$ is commutative if $x\ast y=y\ast x$ for all $x,y\in X$.

¶ An element $e\in X$ is a neutral element of $\ast$ if $e\ast x=x$ and $x\ast e=x$ for every $x\in X$.