Commutativity

A binary operation is commutative if

a * b = b * a

for every pair in its domain. A particular pair may commute even when the operation is not commutative. For example, ordinal addition and ordinal multiplication are not generally commutative.

Table of contents
- Commuting elements Commutativity
- Noncommutativity Commutativity

Commuting elements ( $ab = ba$ )

 0  0

Two elements commute when their two products in opposite orders agree. A particular pair of matrices may commute even when matrix multiplication is not a commutative operation on the whole set of matrices.

Noncommutativity ( $\exists a, b : a * b \neq = b * a$ )

 0  0

A binary operation is noncommutative if some pair of elements fails to commute. Thus reversing their order can change the result. The same operation may still have many commuting elements.