A one-to-one correspondence is described between the set ${\cal J}(m)$ of numerical semigroups with multiplicity $m$ and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in ${\cal J}(m)$ a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of ${\bb N}^{m-1}$. Finally, this result is particularized to the symmetric case.