# 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**.