Commutativity - Definition


Commutativity - Definition

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 <math>a\times b\neq b\times a<math>. 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 circuits 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.

See also