Groupoid

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'.

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.

See also

OD structure