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 | + | 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 | + | 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. | + | #'''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. | + | #'''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 | + | 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.
- 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: {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.