On the generators of a generalized numerical semigroup

Abstract

We give a characterization on the sets A ⊆ ℕd such that the monoid generated by A is a generalized numerical semigroup (GNS) in ℕd. Furthermore we give a procedure to compute the hole set ℕd \ S, where S is a GNS, if a finite set of generators of S is known.

Algorithms for generalized numerical semigroups

“…Also generalized numerical semigroups in N d , with d ≥ 2, admit an unique finite minimal set of generators (see [5]; this is true in general in any cancellative monoid, [13]). The most important differences between numerical semigroups and generalized numerical semigroups consist in the identification of a Frobenius element and those generators larger than this element: N has a natural total order while N d has only a natural partial order (that is induced by the total order in N).…”

“…An algorithm to compute the set of gaps of a generalized numerical semigroup from a generating set Let S ⊆ N d be a generalized numerical semigroup and suppose that a finite set of (not necessarily minimal) generators is known. A possible way to compute H(S ) is via the following characterization that appears in [5]. (1) For all j ∈ {1, 2, .…”

“…In [11], the number N g,d is computed for several (low) values, and the concept of generalized numerical semigroup is extended to affine semigroups having finite complement in their associated spanning cones. In [5] (finite) sets generating generalized numerical semigroups are characterized, and a procedure to compute the set of gaps is presented.…”

A generalization of Wilf’s conjecture for generalized numerical semigroups

“…We can identify S * with the monomials in I and S * + S * with the monomials in I 2 . It follows that S * \ (S * + S * ) can be identified with the monomials in I but not in (1,4), (1,5), (1,6), (2, 2), (2, 3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (6, 1)}; these are the exponent vectors of monomials in I but not in I 2 . Figure 2 provides a graphical view of this GNS -black points are the holes of the GNS, while the red points are the minimal generators.…”

Irreducible generalized numerical semigroups and uniqueness of the Frobenius element

“…A monoid S ⊆ N d is called a generalized numerical semigroup (GNS) if H(S) = N d \S is a finite set. As for numerical semigroups, the elements of H(S) are called the holes of S and the cardinality g = |H(S)| is called the genus of S. In [1,3] several ideas originating in numerical semigroups were extended to generalized numerical semigroups and several new tools were introduced in order to handle the differences. A crucial difference between numerical semigroups in N and generalized numerical semigroups in N d is the definition of the multiplicity and the Frobenius element of S. There is a natural ordering of the elements of N that respects the monoid operation of addition.…”