4.5.1.26: Groupoid

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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).