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
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 $\displaystyle e$ be a left identity and $\displaystyle x \in S.$ so $\displaystyle \exists y \in S: \ yx=e.$ also $\displaystyle \exists z \in S: \ zy=e.$ thus $\displaystyle e=zy=z(ey)=z(yx)y=(zy)xy=e(xy)=xy.$
so $\displaystyle y$ is also a right inverse of $\displaystyle x.$ finally we have: $\displaystyle xe=x(yx)=(xy)x=ex=x,$ i.e. $\displaystyle e$ is a right identity as well. Q.E.D.