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In [[mathematics]], an '''antihomomorphism''' is a type of [[Function (mathematics)|function]] defined on sets with multiplication that reverses the order of multiplication. An '''antiautomorphism''' is an antihomomorphism that is a [[bijection]] from an object to itself.
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<sup>-1</sup>'' 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 if φ : ''X'' → ''Y'' is a ring homomorphism,
:φ(''x+y'') = φ(''x'')+φ(''y'')
:φ(''xy'') = φ(''y'')φ(''x'')
for all ''x,y'' in ''X''. For [[algebra over a field|algebras over a field]] ''K'', φ must be a ''K''-[[linear map]] of the underlying [[vector space]].
The operation of matrix [[transpose]] is an example of a ring antiautomorphism.
Note that if the multiplication in the range of φ is [[commutative]], then an antihomomorphism is the same thing as a [[homomorphism]] and an antiautomorphism is the same thing as an [[automorphism]].
One can also define an antihomomorphism as an homomorphism from ''X'' to the opposite object ''Y''<sup>op</sup> (which is identical to ''X'' but with multiplication reversed).
The [[function composition|composition]] of two antihomomorphisms is always an homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with an automorphism gives another antiautomorphism.
It is frequently the case that antiautomorphisms are [[involution]]s, i.e. the square of the antiautomorphism is the [[identity map]].
==Examples==
* The map that sends ''x'' to its [[inverse element|inverse]] ''x''<sup>−1</sup> is an antiautomorphism in any group.
* The [[transpose]] map (or [[conjugate transpose]] map) is an antiautomorphism on the algebra of square [[matrix (mathematics)|matrices]].
* The [[Hermitian adjoint]] is an antiautomorphism on the algebra of [[linear operator]]s on a [[Hilbert space]].
* More generally, the *-involution in any [[star-algebra]] is an antiautomorphism.
* The conjugation involution in any [[Cayley-Dickson algebra]] such as the [[quaternion]]s and [[octonion]]s.
[[Category:Abstract algebra]]
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