Groupoid

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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 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 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 & 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