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Math Help - Binary operations, groupoids

  1. #1
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    Jan 2010
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    Question Binary operations, groupoids

    I have the following problem:

    Let (M; *) be a groupoid in which the following relations hold:
    i) x * x = x; x belongs to M
    ii) (x * y) * z = (y * z) * x; x, y, z belong to M.
    Show that the operation * is commutative.

    I don't know how to start to prove this. Please help Much appreciated!
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  2. #2
    Member Black's Avatar
    Joined
    Nov 2009
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    105
    Let x,y be in M. Then

    x*y = x*(y*y) (using (i))
    = (x*y)*y (associativity)
    = (y*y)*x (using (ii))
    = y*x (using (i).
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