A binary operation on a set $M$ is a map from the set of ordered pairs $M^2=\{ (a,b) | $ $ a \in M, b \in M\}$ into $M$. I.e. it is a way to establish a correspondence between a pair of elements from $M$ and an element from $M$. Image of pair $(a,b)$ is denoted by $a \cdot b $.