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Difference between revisions of "Complex"

From Online Dictionary of Crystallography

 
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<font color="blue">Complexe</font> (''Fr''). <font color="red">Komplex</font> (''Ge''). <font color="black">Complesso</font> (''It''). <font color="green">Complejo</font> (''Sp'').
 
==Definition==
 
==Definition==
A '''complex''' is a subset obtained from a group by chosing part of its elements in such a way that the closure property of groups is not respected. Therefore, a complex is not a group itself.
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A '''complex''' is a subset obtained from a group by choosing part of its elements in such a way that the closure property of groups is not respected. Therefore, a complex is not a group itself.
  
 
A typical example of complexes is that of [[coset]]s. In fact, a coset does not contain the identity and therefore it is not a group.
 
A typical example of complexes is that of [[coset]]s. In fact, a coset does not contain the identity and therefore it is not a group.
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==Laws of composition for complexes==
 
==Laws of composition for complexes==
There exist two laws of compositions for complexes.
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There exist two laws of composition for complexes.
#'''Addition'''. The sum of two complexes K and L consists of all the elements of K and L combined. The addition of complexes is therefore a union from a set-theoretic viewpoint. It is commutative and associative.
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#'''Addition'''. The sum of two complexes ''K'' and ''L'' consists of all the elements of ''K'' and ''L'' combined. The addition of complexes is therefore a union from a set-theoretic viewpoint. It is commutative and associative.
#'''Multiplication'''. The product of two complexes K and L is the complex obtained by formal expansion: {K<sub>i</sub>L<sub>j</sub>}. It is, in general, non-commutative, but associative and distributive.
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#'''Multiplication'''. The product of two complexes ''K'' and ''L'' is the complex obtained by formal expansion: {''K<sub>i</sub>L<sub>j</sub>''}. It is, in general, non-commutative, but associative and distributive.
  
It is, in general, not permissible to apply the cancelling rule to complexes. This means that from the equation KL = KM does '''not''' follow that: L = M, unless K is a single element.
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It is, in general, not permissible to apply the cancelling rule to complexes. This means that from the equation ''KL = KM'' does '''not''' follow that ''L = M'', unless ''K'' is a single element.
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 16:27, 18 June 2019

Complexe (Fr). Komplex (Ge). Complesso (It). Complejo (Sp).

Definition

A complex is a subset obtained from a group by choosing part of its elements in such a way that the closure property of groups is not respected. Therefore, a complex is not a group itself.

A typical example of complexes is that of cosets. In fact, a coset does not contain the identity and therefore it is not a group.

A subgroup is a particular case of complex that obeys the closure property and is a group itself.

Laws of composition for complexes

There exist two laws of composition for complexes.

  1. Addition. The sum of two complexes K and L consists of all the elements of K and L combined. The addition of complexes is therefore a union from a set-theoretic viewpoint. It is commutative and associative.
  2. Multiplication. The product of two complexes K and L is the complex obtained by formal expansion: {KiLj}. It is, in general, non-commutative, but associative and distributive.

It is, in general, not permissible to apply the cancelling rule to complexes. This means that from the equation KL = KM does not follow that L = M, unless K is a single element.