The Covariety of Saturated Numerical Semigroups with Fixed Frobenius Number

Abstract

In this work, we show that if F is a positive integer, then Sat(F) = {S | S is a saturated numerical semigroup with Frobenius number F} is a covariety. As a consequence, we present two algorithms: one that computes Sat(F), and another which computes all the elements of Sat(F) with a fixed genus. If X ⊆ S\Δ(F) for some S ∈ Sat(F), then we see that there exists the least element of Sat(F) containing X. This element is denoted by Sat(F)[X]. If S ∈ Sat(F), then we define the Sat(F)-rank of S as the minimum of {cardinality(X) | S = Sat(F)[X]}. In this paper, we present an algorithm to compute all the elements of Sat(F) with a given Sat(F)-rank.