1.51: Groupoid
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 '.
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).