Antihomomorphism

In mathematics , an anti-homomorphism is a function that is defined on two sets , each with a two-digit link , and that reverses the order of the operands.An anti-isomorphism is a bijective anti-homomorphism.An anti-endomorphism is an anti-homomorphism in which the definition set and the target set match. An anti- automorphism is an anti-isomorphism that is also an anti-endomorphism.

Formal definition

Be and quantities on which a calculation rule or two-digit link , e.g. B. a multiplication, ${\displaystyle D\,}$ ${\displaystyle Z\,}$

${\displaystyle d_{1}\times _{\!D}d_{2}=d\,}$      and      ${\displaystyle z_{1}\times _{\!Z}z_{2}=z\,}$

exists and is

${\displaystyle \phi \colon D\to Z}$

a mapping between the two sets. Then antihomomorphism is called if ${\displaystyle \phi \,}$

${\displaystyle \phi (d_{1}\times _{\!D}d_{2})=\phi (d_{2})\times _{\!Z}\phi (d_{1})\,}$

is. In contrast to homomorphism , antihomomorphism reverses the factors in the target set.

Examples

1. In the group theory is the inverse image with a antiautomorphism.
         ${\displaystyle \phi \colon G\to G\qquad }$ ${\displaystyle \qquad \phi (g)=g^{-1}\,}$
   
2. In ring theory , an antihomomorphism is a mapping between two rings, which reverses the order during multiplication, while this does not play a role in addition, which is commutative anyway. An important example is the transposition of a matrix
         ${\displaystyle {\begin{aligned}(A+B)^{T}&=A^{T}+B^{T}\\(A\cdot B)^{T}&=B^{T}\cdot A^{T}\\\end{aligned}}}$
3. Another example of a ring antihomomorphism is the conjugation of the quaternions :
         ${\displaystyle {\begin{aligned}{\overline {x+y}}&={\bar {x}}+{\bar {y}}\\{\overline {x\cdot y}}&={\bar {y}}\cdot {\bar {x}}\\\end{aligned}}}$
4. If G is a group and an automorphism, then it is an anti- automorphism . ${\displaystyle \psi \colon G\to G}$ ${\displaystyle g\mapsto \psi (g^{-1})}$

Involutive anti-automorphism

The first 3 of the above anti-automorphisms are also involutions , i.e. H. the duplicate results in the identical figure . With the terms above, the following applies:

1. ${\displaystyle (g^{-1})^{-1}=g\,}$
2. ${\displaystyle \left(A^{T}\right)^{T}=A}$
3. ${\displaystyle {\overline {({\bar {x}})}}=x}$.

One then speaks of an involutive anti-automorphism. Occasionally there is also the somewhat shortened term “anti-involution”.

The antiautomorphism in the last example is only involutive if the automorphism itself is already involutive. ${\displaystyle \psi \,}$

comment

In the case of an antihomomorphism (and an antiisomorphism), either in the definition set or in the target set, the link, if there is no further reference to it, can be replaced by a third , say: ${\displaystyle \times \,}$

${\displaystyle z_{2}\times _{\!Z}z_{1}=:z_{1}\times z_{2}\,}$.

Such a redefinition turns the antihomomorphism into a homomorphism in the new link.

In the case of anti-endomorphisms (and anti-auto-morphisms), however, the reference is duplicated from the start, since the link in the definition set and the target set is the same. Here nothing is gained by a redefinition.

Other properties

If the connection of the target set is commutative , then an antihomomorphism is the same as a homomorphism.

The combination of two antihomomorphisms results in a homomorphism. The composition of an antihomomorphism with a homomorphism results in an antihomomorphism.

See also

• Counter ring

Web links

• Eric W. Weisstein : Antiautomorphism . In: MathWorld (English).