Numerical Semigroups Where Certain Sequences of Small Elements Are Forbidden

Let a1,a2,…,ap be integers such that 1≤a1<a2<⋯<ap. A (a1,…,ap)-semigroup is a numerical semigroup S satisfying that if {x,x+a1,…,x+ap}⊆S, then its Frobenius number is less than x+ap. In this work we will study this class of numerical semigroups. Our goal is to show that they form a Frobenius variety. Similarly, those with a given multiplicity or a fixed Frobenius number constitute a Frobenius pseudo-variety or a covariety, respectively. Exploiting the fact that these sets can be organized into trees, we have developed algorithms to calculate their members.