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 $\mathbb U$:


 * $\circ: S \times T \to \mathbb U: \circ \tuple {s, t} = y \in \mathbb U$

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

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.

uses $\bigtriangleup$ and $\bigtriangledown$ for the general binary operation, which has the advantage that they are unlikely to be confused with anything else in this context.

uses $\intercal$, and calls it truc, French for trick or technique:


 * The symbol $\intercal$ is called truc ("trook") and is French for "thingummyjig"! The idea it conveys is that what we call our law of composition does not matter, for what we are really interested in are sets of objects and mappings between them.

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,

Also see

 * Binary Operation on Subset is Binary Operation
 * Definition:Iterated Binary Operation