Definition:Operation/Binary Operation

Definition
A binary operation is the special case of an operation where the operation has exactly two operands.

A binary operation is a mapping $\circ$ from the Cartesian product of two sets $S \times T$ to a universe $\mathbb U$:


 * $\circ: S \times T \to \mathbb U: \circ \left ({s, t}\right) = y \in \mathbb U$

If $S = T$, then $\circ$ can be referred to as a binary operation on $S$.

Note that a binary operation is a special case of a general operator, i.e. one that has two operands.

If $\circ$ is a binary operation on $S$, then for any $T \subseteq S$, $\circ \left ({x, y}\right)$ is defined for every $x, y \in T$.

So $\circ$ is a binary operation on every $T \subseteq S$.

Also known as
Some authors use the term (binary) composition or law of composition for (binary) operation.

Most authors use $\circ$ for composition of relations (which, if you think about it, is itself an operation) as well as for a general operation. To avoid confusion, some authors use $\bullet$ for composition of relations to avoid ambiguity.

Some authors use $\intercal$ (or a variant) called truc (pronounced trook, French for trick or technique ).

Also defined as
Some authors specify that a binary operation $\circ$ is defined such that the codomain of $\circ$ is the same underlying set as that which forms the domain.

That is:
 * $\circ: S \times S \to S$

and thus gloss over the fact that a binary operation defined in such a way is closed.

Such a treatment can obscure the detail of the development of the theory of algebraic substructures; closedness is a fundamental concept in this context,