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.
R-varieties Frobenius restricted number Frobenius varieties Frobenius pseudo-varieties Monoids Numerical semigroups Tree (associated to an R-variety)
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The authors would like to thank both of the referees for providing constructive comments and help in improving the contents of this paper.