Canonical commutation relation: Difference between revisions

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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.
 
==Definition==
Informally, an antihomomorphism is map that switches the order of multiplication.
 
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.)
 
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]]).
 
==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''<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.
 
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))
 
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''.<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>
 
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.
 
===Involutions===
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'''.
 
* The map that sends ''x'' to its [[inverse element|inverse]] ''x''<sup>−1</sup> is an involutive antiautomorphism in any group.
 
A ring with an involutive antiautomorphism is called a [[*-ring]], and [[*-algebra#Examples|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 [[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.
 
==See also==
* [[Semigroup with involution]]
 
==References==
{{reflist}}
*{{MathWorld|title=Antihomomorphism|urlname=Antihomomorphism}}
 
[[Category:Morphisms]]

Revision as of 01:33, 2 March 2014

Making a computer run quick is very simple. Most computers run slow because they are jammed up with junk files, that Windows has to look through each time it wants to find anything. Imagine having to discover a book inside a library, yet all of the library books are in a big big pile. That's what it's like for the computer to find anything, whenever your program is full of junk files.

StreamCI.dll errors are caused by a number of different difficulties, including which the file itself has been moved on the program, the file is outdated or you have installed certain third-party sound motorists that are conflicting with the file. The advantageous news is the fact that if you need to resolve the error you're seeing, you need to look to initially ensure the file & motorists are functioning okay on your PC and also then resolving any StreamCI.dll errors that may be inside the registry of the computer.

Naturally, the next logical step is to receive these false entries cleaned out. Fortunately, this might be not a difficult task. It is the second thing you should do when you observed your computer has lost speed. The first will be to create sure there are no viruses or severe spyware present.

Fixing tcpip.sys blue screen is easy to do with registry repair software.Trying to fix windows blue screen error on the own may be challenging because in the event you remove or damage the registry it will cause serious damage to your computer. The registry must be cleaned plus all erroneous plus incomplete info removed to stop blue screen errors from occurring.The benefit of registry repair software is not limited to only getting rid of the blue screen on startup.We may be amazed at the better plus more improved speed and performance of your computer program following registry cleaning is done. Registry cleaning could really develop a computer's working abilities, especially when we choose a certain registry repair software which is especially efficient.

To fix the problem which is caused by registry error, you have to utilize a tuneup utilities. That is the safest and easiest means for average PC users. However there are thousands of registry cleaners accessible out there. You have to find a wise one which may really solve a issue. If you use a terrible 1, we may expect more difficulties.

The most probable cause of the trouble is the system problem - Registry Errors! That is the reason why persons that already have more than 2 G RAM on their computers are nonetheless frequently bothered by the problem.

Another problem with the cracked variation is that it takes too much time to scan the program plus whilst it is scanning, you can not employ the computer otherwise. Moreover, there is no technical help to these damaged versions meaning should you get stuck somewhere, you can't ask for help. They even do not have any customer service help lines wherein we could call or send to resolve your issues.

So, the number one thing to do whenever the computer runs slow is to buy an authentic plus legal registry repair tool which would assist we eliminate all difficulties associated to registry and assist we enjoy a smooth running computer.

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