# Groupoid

### From Online Dictionary of Crystallography

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 e
_{x}(left unit of x), e_{x}' (right unit of x) and x^{-1}("inverse" of x) such that:- e
_{x}*x = x - x* e
_{x}' = x - x
^{-1}*x = e_{x}'.

- e

From these properties it follows that:

- x* x
^{-1}= e_{x},*i*.*e*. that that e_{x}is right unit for x^{-1}, - e
_{x}' is left unit for x^{-1} - e
_{x}and e_{x}' are idempotents,*i*.*e*. e_{x}* e_{x}= e_{x}and e_{x}'* e_{x}' = e_{x}'.

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.