# Semi-Groups

• Feb 22nd 2006, 04:18 AM
topsquark
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
• Feb 22nd 2006, 02:55 PM
ThePerfectHacker
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 \$\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.