Binary Operation on a Set

Let $S$ be a set. A binary operation on $S$ is a function

where $S\times S$ denotes the Cartesian product of $S$ with itself. For $a,b\in S$ we write $a\ast b$ for the value of $\ast$ at the pair $(a,b)$.

A binary operation $\ast$ on $S$ is called:

1. associative if $(a\ast b)\ast c=a\ast(b\ast c)$ for all $a,b,c\in S$;
2. commutative if $a\ast b=b\ast a$ for all $a,b\in S$.

A subset $T\subseteq S$ is said to be closed under $\ast$ if $a\ast b\in T$ for all $a,b\in T$. For such a $T$, the restriction of $\ast$ to $T$ is the binary operation on $T$ given by $(a,b)\mapsto a\ast b$ for $a,b\in T$.