Actions

Difference between revisions of "Groupoid"

From Online Dictionary of Crystallography

 
(6 intermediate revisions by 2 users not shown)
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="black">Gruppoide</font> (''It''). <font color="purple">亜群</font> (''Ja''). <font color="brown">Группоид</font> (''Ru''). <font color="green">Grupoide</font> (''Sp'').  
  
  
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>&minus;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>&minus;1</sup> * ''x'' = ''e''<sub>''x''</sub>'.
  
 
From these properties it follows that:
 
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>&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>'.
+
* ''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 operations act but neither the identity nor the inversion are included. For this second meaning nowadays the term '''magma''' is used instead (Bourbaki, 1998).
+
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).
  
 
==References==
 
==References==
*Bourbaki, N. (1998) ''Elements of Mathematics: Algebra 1''. Springer.
+
*Bourbaki, N. (1998). ''Elements of Mathematics: Algebra 1''. Springer.
*Brandt H (1927) ''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==
 
==See also==
[[OD structure]]
+
*[[OD structure]]
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 13:37, 28 February 2018

Groupoïde (Fr). Gruppoid (Ge). Gruppoide (It). 亜群 (Ja). Группоид (Ru). Grupoide (Sp).


A groupoid (G,*) is a set G with a law of composition * mapping of a subset of G × G into G. The properties of a groupoid are:

  • if x, y, zG 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 yG are such that x*y and x' * y are defined and equal, then x = x' (cancellation property);
  • for all xG 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 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'.

A groupoid can be seen as:

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 operations 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.
  • Brandt, H. (1927). Mathematische Annalen, 96, 360-366.
  • Hausmann, B. A. and Ore, O. (1937). American Journal of Mathematics, 59, 983-1004.

See also