Binary operations/Introduction/Section

Definition

An operation (or binary operation) ${}\circ$ on a set ${}M$ is a mapping

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto \circ (x,y)=x\circ y.$

A binary operation assigns to a pair

${}(x,y)\in M\times M\,$

another element

${}x\circ y\in M\,.$

Many mathematical constructions are captured by this concept: addition, difference, multiplication, division of numbers, the composition of mappings, the intersection or the union of sets, etc. Basically, any symbol can be used to denote a binary operation, like ${}\circ ,\cdot ,+,-,\oplus ,\clubsuit ,\heartsuit$. Depending on the symbol, we call the binary operation also multiplication or addition, but this does not mean that we are referring to any natural multiplication. Important structural properties of a binary operation are listed in the following definitions.

Definition

A binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

on a set ${}M$ is called commutative if for all ${}x,y\in M$ the equality

${}x\circ y=y\circ x\,$

holds.

Definition

A binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

on a set ${}M$ is called associative if for all ${}x,y,z\in M$ the equality

${}(x\circ y)\circ z=y\circ (x\circ z)\,$

holds.

Definition

Let a set ${}M$ and a binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

be given. An element ${}e\in M$ is called neutral element of the operation if, for all ${}x\in M$, the equalities ${}x\circ e=x=e\circ x$

hold.

In the commutative case, it is enough to check only one property of the neutral element.

Definition

Let a set ${}M$ with a binary operation

$\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto x\circ y,$

and a neutral element ${}e\in M$ be given. For an element ${}x\in M$, an element ${}y\in M$ is called inverse element (for ${}x$) if the equalities

${}x\circ y=e=y\circ x\,$

hold.