If a seimigroup's left identity is unique, can it be two-side identity? 
If a seimigroup's left identity is unique, can it be two-side identity?  

The answer is true if we talk it in a ring. Like construct $(be-b+e)a=a$. But in a semigroup I can't image how to construct a equation so we can use the condition uniqueness. So I am wondeting if the statement is false. But I also couldn't get a counterexample.

That is an interesting question.
Let $S=${$x,y$} (where $x$ is not equal with $y$). $S$ with the following operation is a semigroup:
$xy=y ,xx=y, yx=y, yy=y$.
We see that $y$ is the unique left identity of $S$, but $y$ is not an identity, other wise we have $yx=x$ then we have $x=y$.
