Definition:Commutative/Operation

Definition

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

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

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

That is, if every pair of elements of $S$ commutes.

Also known as

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

Also see

• Results about commutativity can be found here.

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.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next). Commutative and associative operations: $\S 4.2 \ (1)$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.5$
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Operations
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 28$. Associativity and commutativity: Definition $2$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): commutative or permutable: 1. (of an operator)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): commutative
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): commutative
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): commutative
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): commutative