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Groupoid

  • Page ID
    18984
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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 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).


    Groupoid is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Online Dictionary of Crystallography.

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