TITLE: Semigroups and solutions of equation
QUESTION [11 upvotes]: It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ya=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ya=b$ has solution(not necessarily unique given) would it be a group?

REPLY [2 votes]: Here is a short self-contained proof.
Suppose $S$ is a semigroup (finite or infinite, but of course nonempty) such that, for any $a,b\in S$, each of the equations $ax=b$ and $ya=b$ has at least one solution in $S$. We will show that $S$ is a group. (In fact this is how groups are defined in some textbooks. If memory serves, this is how they are defined in a book by Kurosh.)
Choose elements $a\in S$ and $e\in S$ such that $ae=a$.
Consider any $x\in S$. Choose $y\in S$ so that $ya=x$. Then
$$xe=yae=ya=x.$$
Thus $e$ is a right identity element.
For any element $x\in S$ there is an element $x'\in S$ such that $xx'=e$, and an element $x''\in S$ such that $x'x''=e$. Then
$$x'x=x'xe=x'xx'x''=x'ex''=x'x''=e,$$
that is, $x'$ is a two-sided inverse of $x$ with respect to $e$.
Finally, for any $x\in S$ we have
$$ex=xx'x=xe=x,$$
so $e$ is a two-sided identity element. This completes the proof that $S$ is a group.