“Antihomomorphism”的意思、由来-开放百科全书

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e., an antiisomorphism, from a set to itself. From being bijective it follows that it has an inverse, and that the inverse is also an antiautomorphism.

Definition

Informally, an antihomomorphism is a map that switches the order of multiplication.

Formally, an antihomomorphism between objects

and

is a homomorphism

, where

equals

as a set, but has its multiplication reversed to that defined on

: denoting the, generally non-commutative, multiplication on

with "

", then the multiplication on

denoted as

, is defined by

. The object

is called the opposite object to Y. (Respectively, opposite group, opposite algebra, opposite category etc.)

This definition is equivalent to a homomorphism

(reversing the operation before or after applying the map is equivalent). Formally, sending

and acting as the identity on maps is a functor (indeed, an involution).

Examples

In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if {{nowrap|φ : X → Y}} is a group antihomomorphism,

The map that sends x to x⁻¹ is an example of a group antiautomorphism. Another important example is the transpose operation in linear algebra which takes row vectors to column vectors. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.

With matrices, an example of an antiautomorphism is given by the transpose map. Since inversion and transposing both give antiautomorphisms, their composition is an automorphism. This involution is often called the contragredient map, and it provides an example of an outer automorphism of the general linear group {{nowrap|GL(n, F)}}, where F is a field, except when {{nowrap|1={{abs|F}} = 2}} and {{nowrap|1=n = 1 or 2}} or {{nowrap|1={{abs|F}} = 3}} and {{nowrap|1=n = 1}} (i.e., for the groups {{nowrap|GL(1, 2)}}, {{nowrap|GL(2, 2)}}, and {{nowrap|GL(1, 3)}}).

In ring theory, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So {{nowrap|φ : X → Y}} is a ring antihomomorphism if and only if:

For algebras over a field K, φ must be a K-linear map of the underlying vector space. If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below.

Involutions

It is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphism is the identity map; these are also called {{visible anchor|involutive antiautomorphism}}s.

A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.

Properties

If the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism and an antiautomorphism is the same thing as an automorphism.

The composition of two antihomomorphisms is always a homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with a homomorphism gives another antihomomorphism.

Definition

Examples

Involutions

Properties

See also

References