Definition:Commutative

Definition

Let $\circ$ be a binary operation.

Two elements $x, y$ are said to commute (or permute) iff:

$x \circ y = y \circ x$

The binary operation $\circ$ is commutative on $S$ iff:

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

History

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.

Pronunciation

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

Sources

• Walter Ledermann: Introduction to the Theory of Finite Groups (1949): $\S 2$: Definition $3$
• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
• W.E. Deskins: Abstract Algebra (1964): $\S 1.4$: Definition $1.11 \ \text{(a)}$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.2$
• Seth Warner: Modern Algebra (1965): $\S 2$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.5$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.3$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 28$