Groupoid

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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, zG 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 yG are such that x*y and x' * y are defined and equal, then x = x' (cancellation property);
  • for all xG there exist elements ex (left unit of x), ex' (right unit of x) and x−1 ('inverse' of x) such that:
    • ex * x = x
    • x * ex' = x
    • x−1 * x = ex'.

From these properties it follows that:

  • x * x−1 = ex, i.e. that ex is right unit for x−1,
  • ex' is left unit for x−1
  • ex and ex' are idempotents, i.e. ex * ex = ex and ex' * ex' = ex'.

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