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What is a groupoid?

  1. Jun 28, 2011 #1
    What is a groupoid? I'm reading a book about gyrovectors and in the first chapter it starts defining something which it calls it a groupoid but doesn't inform the reader about its axioms. so I felt uncomfortable to face a new mathematical definition without knowing the axioms that it should satisfy.
    Here is what the book says:

    What axioms should a groupoid satisfy? and what's the difference between a groupoid and group? because the name apparently has been derived from the word group.
  2. jcsd
  3. Jun 28, 2011 #2
    Hi AdrianZ! :smile:

    A groupoid doesn't have any axioms. It's just any set with any binary operation.
    So [itex](\mathbb{N},+)[/itex], [itex](\mathbb{N},\cdot)[/itex], [itex](\mathbb{Z},\cdot)[/itex], etc are all groupoids.

    The only requirement is that the operation must really be an operation. For example [itex](\mathbb{R},/)[/itex] isn't a groupoid as 1/0 doesn't exist.

    A more conventional name for a groupoid is a magma: http://en.wikipedia.org/wiki/Magma_(algebra)
    The name magma is preferred, because groupoid often has another meaning in category theory and Lie algebras.
  4. Jun 28, 2011 #3
    So you mean the binary operation must be well-defined for any two elements in a groupoid? right?
  5. Jun 28, 2011 #4
    Yes, the operation must be well-defined and the set must be closed under the operation. (for example [itex](\mathbb{N},-)[/itex] is not a groupoid since 2-3 is not a natural number. That are the only requirements.

    In other words, the operation

    [tex]*:M\times M\rightarrow M[/tex]

    must be a function.
  6. Jun 28, 2011 #5
    Okay. Thanks.
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