Frobenius restricted varieties in numerical semigroups

The common behaviour of many families of numerical semigroups led to defining, firstly, the Frobenius varieties and, secondly, the (Frobenius) pseudo-varieties. However, some interesting families are still not covered by these definitions. To overcome this situation, here we introduce the concept of Frobenius restricted variety (or R -variety). We generalize most of the results for varieties and pseudo-varieties to R -varieties. In particular, we study the tree structure that arises within them.