Difference between revisions of "Groupoid"
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| − | <font color="blue">Groupoïde</font> (''Fr''). <font color="red">Gruppoid</font> (''Ge''). <font color=" | + | <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 | + | 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, 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 | + | * 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' | + | * 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 e<sub>x</sub> (left unit of x), e<sub>x</sub>' (right unit of x) and x<sup> | + | *for all ''x'' ∈ ''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> | + | ** ''x''<sup>−1</sup> * ''x'' = ''e''<sub>''x''</sub>'. |
From these properties it follows that: | From these properties it follows that: | ||
| − | *x* x<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> | + | * ''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''<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 | + | 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, 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 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:
- a group with a 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 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.