Groupoid - Revision history

MassimoNespolo at 13:37, 28 February 2018

2018-02-28T13:37:54Z

BrianMcMahon: Tidied translations.

2017-11-13T14:46:43Z

Tidied translations.

MassimoNespolo: lang

2017-10-12T08:40:38Z

lang

BrianMcMahon at 12:43, 15 May 2017

2017-05-15T12:43:52Z

BrianMcMahon: Style edits to align with printed edition

2017-05-15T12:41:32Z

Style edits to align with printed edition

MassimoNespolo: /* References */ link

2009-02-21T22:02:19Z

‎References: link

MassimoNespolo: link

2008-12-21T18:33:34Z

link

MassimoNespolo at 09:54, 13 May 2007

2007-05-13T09:54:54Z

MassimoNespolo at 09:26, 28 February 2007

2007-02-28T09:26:30Z

MassimoNespolo at 14:35, 27 February 2007

2007-02-27T14:35:52Z

New page

Grupoïde　(''Fr''); Gruppoid　(''Ge''); Grupoide　(''Sp''); Gruppoide　(''It''); 亜群　(''Ja'').

A '''groupoid''' (G,*) is a set G with a law of composition * mapping of a subset of G x G into G. The properties of a groupoid are:
* if x, y, z ∈ G and if one of the compositions (x*y)*z or x*(y*z) is defined, so is the other and they are equal; (associativity);
* if x, x' and y ∈ G are such that x*y and x'*y are defined and equal, then x = x'; (cancellation property)
*for all x ∈ G there exist elements ex (left unit of x), ex' (right unit of x) and x-1 ("inverse" of x) such that:
** ex*x = x
** x* ex' = x
** x-1*x = ex'.

From these properties it follows that:
*x* x-1 = ex, ''i''.''e''. that that ex is right unit for x-1,
* ex' is left unit for x-1
* ex and ex' are idempotents, ''i''.''e''. ex* ex = ex and ex'* ex' = ex'.

==See also==
[[OD structure]]

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 13:37, 28 February 2018
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	* ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' are idempotents, ''i.e.'' ''e''<sub>''x''</sub> * ''e''<sub>''x''</sub> = ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' * ''e''<sub>''x''</sub>' = ''e''<sub>''x''</sub>'.		* ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' are idempotents, ''i.e.'' ''e''<sub>''x''</sub> * ''e''<sub>''x''</sub> = ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' * ''e''<sub>''x''</sub>' = ''e''<sub>''x''</sub>'.

		+	A groupoid can be seen as:
		+	*a [[group]] with a [[partial symmetry\|partial function]] replacing the [[binary operation]];
		+	*a category in which every morphism is invertible.
	The concept of groupoid as defined here was introduced by Brandt (1927). An alternative meaning of groupoid was introduced by Hausmann and Ore (1937) as a set on which [[binary operation]]s act but neither the identity nor the inversion are included. For this second meaning nowadays the term '''magma''' is used instead (Bourbaki, 1998).		The concept of groupoid as defined here was introduced by Brandt (1927). An alternative meaning of groupoid was introduced by Hausmann and Ore (1937) as a set on which [[binary operation]]s act but neither the identity nor the inversion are included. For this second meaning nowadays the term '''magma''' is used instead (Bourbaki, 1998).

← Older revision		Revision as of 14:46, 13 November 2017
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−	<font color="blue">Groupoïde</font> (''Fr''); <font color="red">Gruppoid</font> (''Ge''); <font color="black">Gruppoide</font> (''It''); <font color="purple">亜群</font> (''Ja''); <font color="brown">Группоид</font> (''Ru''); <font color="green">Grupoide</font> (''Sp'').	+	<font color="blue">Groupoïde</font> (''Fr''). <font color="red">Gruppoid</font> (''Ge''). <font color="black">Gruppoide</font> (''It''). <font color="purple">亜群</font> (''Ja''). <font color="brown">Группоид</font> (''Ru''). <font color="green">Grupoide</font> (''Sp'').

← Older revision		Revision as of 08:40, 12 October 2017
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−	<font color="blue">Groupoïde</font> (''Fr''). <font color="red">Gruppoid</font> (''Ge''). <font color="~~green~~">~~Grupoide~~</font> (''Sp''). <font color="~~black~~">~~Gruppoide~~</font> (''It''). <font color="~~purple~~">亜群</font> (''Ja'').	+	<font color="blue">Groupoïde</font> (''Fr''); <font color="red">Gruppoid</font> (''Ge''); <font color="black">Gruppoide</font> (''It''); <font color="purple">亜群</font> (''Ja''); <font color="brown">Группоид</font> (''Ru''); <font color="green">Grupoide</font> (''Sp'').

@@ Line 5: / Line 5: @@
 * if ''x'', ''y'', ''z'' &isin; ''G'' and if one of the compositions (''x''*''y'')*''z'' or ''x''*(''y''*''z'') is defined, so is the other and they are equal (associativity);
 * if ''x'', ''x' '' and ''y'' &isin; ''G'' are such that ''x''*''y'' and ''x' ''* ''y'' are defined and equal, then ''x'' = ''x' '' (cancellation property);
-*for all ''x'' &isin; ''G'' there exist elements ''e''<sub>''x''</sub> (left unit of ''x''), ''e''<sub>''x''</sub>' (right unit of ''x'') and ''x''<sup>-1</sup> ('inverse' of ''x'') such that:
+*for all ''x'' &isin; ''G'' there exist elements ''e''<sub>''x''</sub> (left unit of ''x''), ''e''<sub>''x''</sub>' (right unit of ''x'') and ''x''<sup>&minus;1</sup> ('inverse' of ''x'') such that:
 ** ''e''<sub>''x''</sub> * ''x'' = ''x''
 ** ''x'' * ''e''<sub>''x''</sub>' = ''x''
-** ''x''<sup>-1</sup> * ''x'' = ''e''<sub>''x''</sub>'.
+** ''x''<sup>&minus;1</sup> * ''x'' = ''e''<sub>''x''</sub>'.
 From these properties it follows that:
-*''x'' * x<sup>-1</sup> = ''e''<sub>''x''</sub>, ''i.e''. that ''e''<sub>''x''</sub> is right unit for ''x''<sup>-1</sup>,
+*''x'' * x<sup>&minus;1</sup> = ''e''<sub>''x''</sub>, ''i.e''. that ''e''<sub>''x''</sub> is right unit for ''x''<sup>&minus;1</sup>,
-* ''e''<sub>''x''</sub>' is left unit for ''x''<sup>-1</sup>
+* ''e''<sub>''x''</sub>' is left unit for ''x''<sup>&minus;1</sup>
 * ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' are idempotents, ''i.e.'' ''e''<sub>''x''</sub> * ''e''<sub>''x''</sub> = ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' * ''e''<sub>''x''</sub>' = ''e''<sub>''x''</sub>'.

@@ Line 2: / Line 2: @@
-A '''groupoid''' (G,*) is a set G with a law of composition * mapping of a subset of G x G into G. The properties of a groupoid are:
+A '''groupoid''' (''G'',*) is a set ''G'' with a law of composition * mapping of a subset of ''G'' &times; ''G'' into ''G''. The properties of a groupoid are:
-* if x, y, z &isin; G and if one of the compositions (x*y)*z or x*(y*z) is defined, so is the other and they are equal; (associativity);
+* if ''x'', ''y'', ''z'' &isin; ''G'' and if one of the compositions (''x''*''y'')*''z'' or ''x''*(''y''*''z'') is defined, so is the other and they are equal (associativity);
-* if x, x' and y &isin; G are such that x*y and x'*y are defined and equal, then x = x'; (cancellation property)
+* if ''x'', ''x' '' and ''y'' &isin; ''G'' are such that ''x''*''y'' and ''x' ''* ''y'' are defined and equal, then ''x'' = ''x' '' (cancellation property);
-*for all x &isin; G there exist elements e<sub>x</sub> (left unit of x), e<sub>x</sub>' (right unit of x) and x<sup>-1</sup> ("inverse" of x) such that:
+*for all ''x'' &isin; ''G'' there exist elements ''e''<sub>''x''</sub> (left unit of ''x''), ''e''<sub>''x''</sub>' (right unit of ''x'') and ''x''<sup>-1</sup> ('inverse' of ''x'') such that:
-** e<sub>x</sub>*x = x
+** ''e''<sub>''x''</sub> * ''x'' = ''x''
-** x* e<sub>x</sub>' = x
+** ''x'' * ''e''<sub>''x''</sub>' = ''x''
-** x<sup>-1</sup>*x = e<sub>x</sub>'.
+** ''x''<sup>-1</sup> * ''x'' = ''e''<sub>''x''</sub>'.
 From these properties it follows that:
-*x* x<sup>-1</sup> = e<sub>x</sub>, ''i''.''e''. that that e<sub>x</sub> is right unit for x<sup>-1</sup>,
+*''x'' * x<sup>-1</sup> = ''e''<sub>''x''</sub>, ''i.e''. that ''e''<sub>''x''</sub> is right unit for ''x''<sup>-1</sup>,
-* e<sub>x</sub>' is left unit for x<sup>-1</sup>
+* ''e''<sub>''x''</sub>' is left unit for ''x''<sup>-1</sup>
-* e<sub>x</sub> and e<sub>x</sub>' are idempotents, ''i''.''e''. e<sub>x</sub>* e<sub>x</sub> = e<sub>x</sub> and e<sub>x</sub>'* e<sub>x</sub>' = e<sub>x</sub>'.
+* ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' are idempotents, ''i.e.'' ''e''<sub>''x''</sub> * ''e''<sub>''x''</sub> = ''e''<sub>''x''</sub> and ''e''<sub>''x''</sub>' * ''e''<sub>''x''</sub>' = ''e''<sub>''x''</sub>'.
-The concept of groupoid as defined here was introduced by Brandt (1927). An alternative meaning of groupoid was introduced by Hausmann & Ore (1937) as a set on which [[binary operation]]s act but neither the identity nor the inversion are included. For this second meaning nowadays the term '''magma''' is used instead (Bourbaki, 1998).
+The concept of groupoid as defined here was introduced by Brandt (1927). An alternative meaning of groupoid was introduced by Hausmann and Ore (1937) as a set on which [[binary operation]]s act but neither the identity nor the inversion are included. For this second meaning nowadays the term '''magma''' is used instead (Bourbaki, 1998).
 ==References==
-*Bourbaki, N. (1998) ''Elements of Mathematics: Algebra 1''. Springer.
+*Bourbaki, N. (1998). ''Elements of Mathematics: Algebra 1''. Springer.
-*Brandt H (1927) ''[http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27941 Mathematische Annalen]'', '''96''', 360-366.
+*Brandt, H. (1927). ''[http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27941 Mathematische Annalen]'', '''96''', 360-366.
-*Hausmann, B. A. and Ore, O. (1937) ''American Journal of Mathematics'', '''59''', 983-1004.
+*Hausmann, B. A. and Ore, O. (1937). ''American Journal of Mathematics'', '''59''', 983-1004.
 ==See also==
-[[OD structure]]
+*[[OD structure]]
 [[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<font color="blue">Groupoïde</font>　(''Fr''); <font color="red">Gruppoid</font>　(''Ge''); <font color="green">Grupoide</font>　(''Sp''); <font color="black">Gruppoide</font>　(''It''); <font color="purple">亜群</font>　(''Ja'').
+<font color="blue">Groupoïde</font> (''Fr''). <font color="red">Gruppoid</font> (''Ge''). <font color="green">Grupoide</font> (''Sp''). <font color="black">Gruppoide</font> (''It''). <font color="purple">亜群</font> (''Ja'').
@@ Line 14: / Line 14: @@
 * e<sub>x</sub>' is left unit for x<sup>-1</sup>
 * e<sub>x</sub> and e<sub>x</sub>' are idempotents, ''i''.''e''. e<sub>x</sub>* e<sub>x</sub> = e<sub>x</sub> and e<sub>x</sub>'* e<sub>x</sub>' = e<sub>x</sub>'.
 ==See also==

← Older revision		Revision as of 09:26, 28 February 2007
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−	<font color="blue">~~Grupoïde~~</font>　(''Fr''); <font color="red">Gruppoid</font>　(''Ge''); <font color="green">Grupoide</font>　(''Sp''); <font color="black">Gruppoide</font>　(''It''); <font color="purple">亜群</font>　(''Ja'').	+	<font color="blue">Groupoïde</font>　(''Fr''); <font color="red">Gruppoid</font>　(''Ge''); <font color="green">Grupoide</font>　(''Sp''); <font color="black">Gruppoide</font>　(''It''); <font color="purple">亜群</font>　(''Ja'').