In the abstract algebra class, we have proved the fact that right identity and right inverse imply a group, while right identity and left inverse do not.
My question: Are there any good examples of sets (with operations on) with right identity and left inverse, not being a group?
To be specific, suppose $(X,\cdot)$ is a set with a binary operation satisfies the following conditions:
(i) $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for any $a,b,c\in X$;
(ii) There exists $e\in X$ such that for every $a\in X$, $a\cdot e=a$;
(iii) For any $a\in X$, there exists $b\in X$ such that $b\cdot a=e$.
I want an example of $(X,\cdot)$ which is not a group.