Groupoid

Groupoïde (Fr); Gruppoid (Ge); Gruppoide (It); 亜群 (Ja); Группоид (Ru); Grupoide (Sp).

A groupoid (G,*) is a set G with a law of composition * mapping of a subset of G × 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 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 and 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