Antihomomorphism

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism which has an inverse as an antihomomorphism; this coincides with it being a bijection from an object to itself.

Definition

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

Formally, an antihomomorphism between "X" and "Y" is a homomorphism $phicolon\; X\; o\; Y^\{mbox\{op$, where $Y^\{mbox\{op$ equals "Y" as a set, but has multiplication reversed: denoting the multiplication on "Y" as $cdot$ and the multiplication on $Y^\{mbox\{op$ as $*$, we have $x*y\; :=\; ycdot\; x$. The object $Y^\{mbox\{op$ is called the opposite object to "Y". (Respectively, opposite group, opposite algebra, etc.)

This definition is equivalent to a homomorphism $phicolon\; X^\{mbox\{op\; o\; Y$ (reversing the operation before or after applying the map is equivalent). Formally, sending "X" to $X^\{mbox\{op$ 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 φ : "X" → "Y" is a group antihomomorphism, :φ("xy") = φ("y")φ("x")for all "x", "y" in "X".

The map that sends "x" to "x"^-1 is an example of a group antiautomorphism.

In ring theory, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So φ : "X" → "Y" is a ring antihomomorphism if and only if::φ(1) = 1:φ("x"+"y") = φ("x")+φ("y"):φ("xy") = φ("y")φ("x")for all "x", "y" in "X".

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 involutive antiautomorphisms.

* The map that sends "x" to its inverse "x"⁻¹ is an involutive antiautomorphism in any group.

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 an automorphism gives another antiautomorphism.