Complex - Revision history

MassimoNespolo: Lang (Fr, It)

2019-06-18T16:27:27Z

Lang (Fr, It)

BrianMcMahon: Added German and Spanish translations (U. Mueller)

2017-11-09T17:13:50Z

Added German and Spanish translations (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-13T10:38:20Z

Style edits to align with printed edition

MassimoNespolo: typo

2013-05-15T15:44:06Z

typo

MassimoNespolo at 17:52, 25 April 2007

2007-04-25T17:52:42Z

New page

==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.

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 [[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 compositions 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: {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.

[[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<font color="red">Komplex</font> (''Ge''). <font color="green">Complejo</font> (''Sp'').
+<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==
 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.

← Older revision		Revision as of 17:13, 9 November 2017
Line 1:		Line 1:
		+	<font color="red">Komplex</font> (''Ge''). <font color="green">Complejo</font> (''Sp'').
	==Definition==		==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 '''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.

@@ Line 7: / Line 7: @@
 ==Laws of composition for complexes==
-There exist two laws of compositions 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.
+#'''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: L = M, unless K is a single element.
+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]]