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| {{Refimprove|date=January 2010}}
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| In [[mathematics]], an '''antihomomorphism''' is a type of [[Function (mathematics)|function]] defined on sets with multiplication that reverses the [[Noncommutative|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.
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| ==Definition==
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| Informally, an antihomomorphism is map that switches the order of multiplication.
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| Formally, an antihomomorphism between ''X'' and ''Y'' is a homomorphism <math>\phi\colon X \to Y^{\text{op}}</math>, where <math>Y^{\text{op}}</math> equals ''Y'' as a set, but has multiplication reversed: denoting the multiplication on ''Y'' as <math>\cdot</math> and the multiplication on <math>Y^{\text{op}}</math> as <math>*</math>, we have <math>x*y := y\cdot x</math>. The object <math>Y^{\text{op}}</math> is called the '''opposite object''' to ''Y''. (Respectively, '''[[opposite group]]''', '''[[opposite algebra]]''', etc.)
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| This definition is equivalent to a homomorphism <math>\phi\colon X^{\text{op}} \to Y</math> (reversing the operation before or after applying the map is equivalent). Formally, sending ''X'' to <math>X^{\text{op}}</math> and acting as the identity on maps is a [[functor]] (indeed, an [[Involution (mathematics)|involution]]).
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| ==Examples==
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| 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,
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| :φ(''xy'') = φ(''y'')φ(''x'')
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| for all ''x'', ''y'' in ''X''.
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| The map that sends ''x'' to ''x''<sup>−1</sup> is an example of a group antiautomorphism. Another important example is the [[transpose]] operation in [[linear algebra]] which takes [[row vector]]s to [[column vector]]s. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.
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| 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 GL(n,F) where F is a field, except when |F|= 2 and n= 1 or 2 or |F| = 3 and n=1 (i.e., for the groups GL(1,2), GL(2,2), and GL(1,3))
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| 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:
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| :φ(1) = 1 | |
| :φ(''x''+''y'') = φ(''x'')+φ(''y'') | |
| :φ(''xy'') = φ(''y'')φ(''x'')
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| for all ''x'', ''y'' in ''X''.<ref>{{cite book | title=The Theory of Rings | series=Mathematical Surveys and Monographs | volume=2 | first=Nathan | last=Jacobson | authorlink=Nathan Jacobson | publisher=[[American Mathematical Society]] | year=1943 | isbn=0821815024 | page=16 }}</ref>
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| For [[algebra over a field|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.
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| ===Involutions===
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| It is frequently the case that antiautomorphisms are [[involution (mathematics)|involution]]s, i.e. the square of the antiautomorphism is the [[identity function|identity map]]; these are also called '''{{visible anchor|involutive antiautomorphism}}s'''.
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| * The map that sends ''x'' to its [[inverse element|inverse]] ''x''<sup>−1</sup> is an involutive antiautomorphism in any group.
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| A ring with an involutive antiautomorphism is called a [[*-ring]], and [[*-algebra#Examples|these form an important class of examples]].
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| ==Properties==
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| 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]].
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| The [[function composition|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.
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| ==See also==
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| * [[Semigroup with involution]]
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| ==References==
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| {{reflist}}
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| *{{MathWorld|title=Antihomomorphism|urlname=Antihomomorphism}}
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| [[Category:Morphisms]]
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