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Math Help - Semigroup with left id and inverse

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    Semigroup with left id and inverse

    Suppose that S is a semigroup with left identity and left inverse, prove that it is a group.

    Well, since it is a semigroup, S already has the associativity property, but how do I show it has the right id and right inverse?

    Thank you
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    Quote Originally Posted by tttcomrader View Post
    Suppose that S is a semigroup with left identity and left inverse, prove that it is a group.

    Well, since it is a semigroup, S already has the associativity property, but how do I show it has the right id and right inverse?

    Thank you
    let e be a left identity and x \in S. so \exists y \in S: \ yx=e. also \exists z \in S: \ zy=e. thus e=zy=z(ey)=z(yx)y=(zy)xy=e(xy)=xy.

    so y is also a right inverse of x. finally we have: xe=x(yx)=(xy)x=ex=x, i.e. e is a right identity as well. Q.E.D.
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