Abstract
We prove that in the ideal case, up to isomorphism, there are only one type of semigroups which are the union of two copies of the free monogenic semigroup. Similarly, there are only five types of semigroups which are the union of three copies of the free monogenic semigroup. And there are only two types of semigroups which are the union of two copies of the free semigroup in two generators. We provide finite presentations for each of these semigroups