Pascal's simplex: Difference between revisions

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In [[Abstract algebra|modern algebra]], an '''antiisomorphism''' (or '''anti-isomorphism''') between [[Mathematical structure|structured sets]] ''A'' and ''B'' is an [[isomorphism]] from ''A'' to the [[Dual (category theory)|opposite]] of ''B'' (or equivalently from the opposite of ''A'' to ''B''). If there exists an antiisomorphism between two structures, they are ''antiisomorphic.''


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Intuitively, to say that two algebraic structures are ''antiisomorphic'' is to say that they are basically opposites of one another.
 
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An example may clarify the idea. Let ''A'' be the [[binary relation set|binary relation]] (or [[Graph (mathematics)|graph]]) consisting of elements {1,2,3} and binary relation <math>\rightarrow</math> defined as follows:
* <math>1 \rightarrow 2;</math>
* <math>1 \rightarrow 3;</math>
* <math>2 \rightarrow 1.</math>


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Let ''B'' be the binary relation set consisting of elements {''a'',''b'',''c''} and binary relation <math>\Rightarrow</math> defined as follows:
 
* <math>b \Rightarrow a;</math>
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* <math>c \Rightarrow a;</math>
 
* <math>a \Rightarrow b.</math>
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Note that the opposite of ''B'' (called ''B''<sup>op</sup>) is the same set of elements with the opposite binary relation <math>\Leftarrow</math>:
* <math>b \Leftarrow a;</math>
* <math>c \Leftarrow a;</math>
* <math>a \Leftarrow b.</math>


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If we replace ''a'', ''b'', and ''c'' with 1, 2, and 3 respectively, we will see that each rule in ''B''<sup>op</sup> is the same as some rule in ''A''. That is, we can define an isomorphism <math>\phi</math> from ''A'' to ''B''<sup>op</sup> by
 
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<math>\phi(n) = \begin{cases}a&\mbox{if }n=1;\\ b&\mbox{if }n=2;\\ c&\mbox{if }n=3.\end{cases}</math>


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This <math>\phi</math> is an antiisomorphism between ''A'' and ''B''.
 
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[[Category:Morphisms]]


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Latest revision as of 12:10, 1 April 2013

In modern algebra, an antiisomorphism (or anti-isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite of A to B). If there exists an antiisomorphism between two structures, they are antiisomorphic.

Intuitively, to say that two algebraic structures are antiisomorphic is to say that they are basically opposites of one another.

An example may clarify the idea. Let A be the binary relation (or graph) consisting of elements {1,2,3} and binary relation defined as follows:

Let B be the binary relation set consisting of elements {a,b,c} and binary relation defined as follows:

Note that the opposite of B (called Bop) is the same set of elements with the opposite binary relation :

If we replace a, b, and c with 1, 2, and 3 respectively, we will see that each rule in Bop is the same as some rule in A. That is, we can define an isomorphism ϕ from A to Bop by

ϕ(n)={aif n=1;bif n=2;cif n=3.

This ϕ is an antiisomorphism between A and B.


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