  In this paper we present the notion of arithmetic variety for numerical
semigroups. We study various aspects related to these varieties such as the
smallest arithmetic that contains a set of numerical semigroups and we exhibit
the root three associated with an arithmetic variety. This tree is not locally
finite; however, if the Frobenius number is fixed, the tree has finitely many
nodes and algorithms can be developed. All algorithms provided in this article
include their (non-debugged) implementation in GAP.
