uniqueness of solutions of $ax=b$ and $ya=b$ in a semigroup .

Question

Suppose $G$ is a semigroup in which every equation of the form $ax=b$ or $ya=b$ has a solution. Does this solution have to be unique?

Answer 1

$x$ has to be $a^{-1}b$ and $y$ has to be $ba^{-1}$. They are unique of course.

Answer 2

This is the counter example @Rebecca noted. In fact, we see that in some cases we don't find any solutions and this happens cause there are more than one solutions in other case.

  gap> a:= Transformation([2, 3,3]);;
       b:= Transformation([ 3,2,3]);;
       s:= Semigroup(a, b);;
       e:=Elements(s);;
       for k in [1..3] do
           for j in [1..3] do
              for i in [1..3] do
              if e[j]*e[i]=e[k] then Print("k:=", k,"   ", "j:=",j, "   ",i,"\n"); 
              fi;
              od; 
           od;
        od;

        k:=1   j:=1   2
        k:=2   j:=2   2
        k:=3   j:=1   1
        k:=3   j:=1   3
        k:=3   j:=2   1
        k:=3   j:=2   3
        k:=3   j:=3   1
        k:=3   j:=3   2
        k:=3   j:=3   3

Answer 3

If a semigroup has a left identity and any element has a left inverse, then it is a group (Hall M.,The theory of groups).

Your semigroup has these properties. Indeed, let $ea=a$ and $ax=b$. Then $eb=eax=b$, so $e$ is a common identity.