# Online Encyclopedia

# Commutative operation

In mathematics, especially abstract algebra, a binary operation * on a set *S* is **commutative** if

*x***y*=*y***x*

for all *x* and *y* in *S*. Otherwise * is **noncommutative**. If

*x***y*=*y***x*

for a particular pair of elements *x* and *y*, then *x* and *y* are said to *commute*.

The most well-known examples of commutative binary operations are addition (a+b) and multiplication (a*b) of real numbers; for example:

- 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
- 2 × 3 = 3 × 2 (since both expressions evaluate to 6)

Among the binary operations that are not commutative are subtraction (*a* − *b*), division (*a*/*b*), exponentiation (*a*^{b}), functional composition (*f*(*g*(*x*))), and tetration (*a*↑↑*b*).

Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. Important non-commutative operations are the multiplication of matrices and the composition of functions.

An abelian group is a group whose operation is commutative.

A ring is a commutative ring if its multiplication is commutative; the addition is commutative in any ring.

In neurophysiology, *commutative* has much the same meaning as in algebra.

Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state:

- In non-commutative algebra, order makes a difference to multiplication, so that . This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuit s that steer the eyes, head and limbs, and sensory circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models.

(Douglas A. Tweed and others, Nature 399, 261 - 263; 20 May 1999). Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex by showing that subjects rotated in darkness can hold their gaze points stable in space---correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.