Algorithms for generalized numerical semigroups

Abstract

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of [Formula: see text] with finite complement in [Formula: see text]. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in [Formula: see text] we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in [Formula: see text] we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). Its implementation allowed us to compute (for various dimensions) the number of numerical semigroups of higher genus than has previously been computed.