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Math Help - Semi-Groups

  1. #1
    Forum Admin topsquark's Avatar
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    Semi-Groups

    I've got one of my own:

    Let G be a semi-group. (That means G is a set with an associative binary operation on it.) Assume further that in G:
    1) There exists a left identity e such that ea=a for all a in G and
    2) For all a in G there exists a right inverse a^{-1} such that aa^{-1}=e.

    The question: Is G a group?

    My answer is "no" but I don't have anything like a proof. What I'm stuck on is the uniqueness of e...I can prove that if there is a d such that da=a for all a in G then d^{-1}=e^{-1}=e, but I can't prove d=e or d is not equal to e...all I have is there doesn't appear to be a contradiction in assuming that more than one left identity exists. What would really be nice is if there were an example showing that G is not a group. (Or, alternately, a proof that it is.)

    Any suggestions? Thanks!

    -Dan
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    Quote Originally Posted by topsquark
    I've got one of my own:

    Let G be a semi-group. (That means G is a set with an associative binary operation on it.) Assume further that in G:
    1) There exists a left identity e such that ea=a for all a in G and
    2) For all a in G there exists a right inverse a^{-1} such that aa^{-1}=e.

    The question: Is G a group?

    My answer is "no" but I don't have anything like a proof. What I'm stuck on is the uniqueness of e...I can prove that if there is a d such that da=a for all a in G then d^{-1}=e^{-1}=e, but I can't prove d=e or d is not equal to e...all I have is there doesn't appear to be a contradiction in assuming that more than one left identity exists. What would really be nice is if there were an example showing that G is not a group. (Or, alternately, a proof that it is.)

    Any suggestions? Thanks!

    -Dan
    G is not a group. Consider this if ex=x for all x implied that ex=xe then why then is it that we define one of the properties of a group such that ex=xe=x? Because there are cases where it is not. I cannot think of a case but semi-groups are hardly ever studied.
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