Difference between revisions of "Groupoid"
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* 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'', ''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); | * 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.'' ''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>'. | ||
Revision as of 12:43, 15 May 2017
Groupoïde (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 × 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'.
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.