What is the difference between a monosemiring and a semigroup?

The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity with respect to the binary operation defined on it.
A non empty set $(S, +, .)$ is said to be a semiring if both the binary operations follow closure and associativity and $(.)$ distributes over $(+)$  from both left and right.
A semiring $(S, +, .)$ is said to be a monosemiring if $x.y=x+y$ for all $x,y$ in $S$. By this definition, we see that in a monosemiring there a common binary operation which is same as that of a semigroup. Moreover, it seems to be possible that every semigroup can be extended to a monosemiring by means of distributivity. What is the exact difference between monosemiring and semigroup?