Difference between revisions of "Groupoid"
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| − | <font color="blue">Groupoïde</font> | + | <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''). |
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* 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>'. | ||
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| + | 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). | ||
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| + | ==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== | ==See also== | ||
Revision as of 09:54, 13 May 2007
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 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'.
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).
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.