Definition:Commutative

Definition

Commuting Elements

Let $\circ$ be a binary operation.

Two elements $x, y$ are said to commute (with each other) if and only if:

$x \circ y = y \circ x$

Commutative Operation

Let $\struct {S, \circ}$ be an algebraic structure.

Then $\circ$ is commutative on $S$ if and only if:

$\forall x, y \in S: x \circ y = y \circ x$

Commuting Set of Elements

Let $\struct {S, \circ}$ be an algebraic structure.

Let $X \subseteq S$ be a subset of $S$ such that:

$\forall a, b \in X: a \circ b = b \circ a$

That is, every element of $X$ commutes with every other element.

Then $X$ is a commuting set of elements of $S$.

Commutative Algebraic Structure

Let $\struct {S, \circ}$ be an algebraic structure whose operation $\circ$ is a commutative operation.

Then $\struct {S, \circ}$ is a commutative (algebraic) structure.

Historical Note

The term commutative was coined by François Servois in $1814$.

Before this time the commutative nature of addition had been taken for granted since at least as far back as ancient Egypt.

Linguistic Note

The word commutative is pronounced with the stress on the second syllable: com-mu-ta-tive.

Also known as

The terms permute and permutable can sometimes be seen instead of commute and commutative.

Also see

• Definition:Commutative Diagram

• Results about commutativity can be found here.