Commutative operation

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For other meanings of commutation, see commutation (disambiguation).

Mathematical meaning

A map or binary operation <math> f:A \times B->C</math> is said to be commutative when, for any x in A and any y in B
<math>f(x,y) = f(y,x)</math>.

For example:

<math>x\times y = y\times x</math>

for all real numbers x and y.

Otherwise, the operation is noncommutative:
<math>x - y = y - x</math>
<math>2x - 2y = 0</math>
<math>2(x-y) = 0</math>
<math>x - y = 0</math>
<math>x = y</math>

So, subtraction is commutative if and only if x = y and noncommutative for any other pair of real numbers.

Additionally, if

<math>x\times y = y\times x</math>

for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers.

The most well-known examples of commutative binary operations are addition and multiplication 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)

Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. In each case, these operations are commutative over their entire domains.

Among the noncommutative binary operations are subtraction (a − b), division (a/b), exponentiation (a^b), function composition (f o g), tetration (a↑↑b), matrix multiplication, and quaternion multiplication.

The subset of the domain on which an operation is commutative is sometimes called the center in algebra.

An abelian group is a group whose group operation is commutative. A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) In a field both addition and multiplication are commutative.

Commutativity can be another name for symmetry. That is, suppose we solve a problem involving parameters x and y, and determine that the solution is equal to <math>f(x,y)</math>. If there exists a subset of values for x and y where the two values can be exchanged without affecting the function, the problem is symmetric. Many symmetries arise naturaly in mathematics out of simpler symmetries, and are commonly found useful for particular kinds of proofs (see WLOG).

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Neurophysiological meaning

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.

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Commutative operation

From Free net encyclopedia

Mathematical meaning

Neurophysiological meaning

See also

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