1. ## 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 $\displaystyle a^{-1}$ such that $\displaystyle aa^{-1}=e$.

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 $\displaystyle d^{-1}=e^{-1}=e$, 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

2. 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 $\displaystyle a^{-1}$ such that $\displaystyle aa^{-1}=e$.

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 $\displaystyle d^{-1}=e^{-1}=e$, 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 $\displaystyle ex=x$ for all $\displaystyle x$ implied that $\displaystyle ex=xe$ then why then is it that we define one of the properties of a group such that $\displaystyle ex=xe=x$? Because there are cases where it is not. I cannot think of a case but semi-groups are hardly ever studied.