Definition:Operation/Binary Operation

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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 universal set $\mathbb U$:

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

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


Infix Notation

Let $\circ: S \times T \to U$ be a binary operation.

When $\map \circ {x, y} = z$, it is common to put the symbol for the operation between the two operands:

$z = x \circ y$

This convention is called infix notation.


Prefix Notation

Let $\circ: S \times T \to U$ be a binary operation.

The convention that places the symbol for the operation before the two operands:

$z = \circ x y$

is called prefix notation.


Postfix Notation

Let $\circ: S \times T \to U$ be a binary operation.

The convention that places the symbol for the operation after the two operands:

$z = x y \mathop \circ$

is called postfix notation.

In infix notation it would be presented as: $z = x \circ y$


Product

Let $z = x \circ y$.

Then $z$ is called the product of $x$ and $y$.

This is an extension of the normal definition of product that is encountered in conventional arithmetic.


Also known as

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

Some sources refer to a binary operation just as an 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.

1965: Seth Warner: Modern Algebra 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.


1975: T.S. Blyth: Set Theory and Abstract Algebra 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.

The $\LaTeX$ code for \(\intercal\) is \intercal .


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


Sources

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