Affine semigroups of maximal projective dimension

On Some Numerical Semigroup Transforms

“…Example 5. Consider the affine semigroups C = (1, 1), (1, 2), (2, 1), (3,1) and S = C \ {(1, 1), (3, 2), (2, 3)} = (1, 2), (2, 1), (2, 2), (3, 1), (3,5) . Let us compute a finite set of generators of S 1 = {λ ∈ N | λg 1 ∈ S}, where g 1 = (1, 1).…”

“…Let us compute a finite set of generators of S 1 = {λ ∈ N | λg 1 ∈ S}, where g 1 = (1, 1). We need to find the non-negative integer solutions of the linear diophantine system Ax = 0, where A has column vectors (1, 2), (2, 1), (2, 2), (3, 1), (3,5), (−1, −1). gap> LoadPackage("num");; gap> NumSgpsUseNormaliz();; [ 6,2,7,3,5 ] gap> Gcd(B); 1 gap> MinimalGenerators(NumericalSemigroup(B)); [ 2,3 ] Therefore, the previous computations allow to check that S 1 = 2, 3 .…”

“…(3) 1 g 3 . In order to compute {k ∈ N | g 1 + kg 3 ∈ S}, with g 1 = (1, 1) and g 3 = (2, 1), we need to find the minimal factorizations of (1, 1) with respect to the set {(1, 2), (2, 1), (2, 2), (3, 1), (3,5), (−2, −1)}. That is, we need to find the set V 13 introduced before.…”

“…gap> LoadPackage("num");; gap> NumSgpsUseNormaliz();; gap> A:=[ [1,2], [2,1], [2,2], [3,1], [3,5],[-2,-1]];; gap> n:=Length(A);; gap> F:=FactorizationsVectorWRTList([1,1], A); [ [ 0, 0, 0, 6, 1, 10 ], [ 0, 0, 1, 1, 0, 2 ], [ 1, 0, 0, 2, 0, 3 ] ] gap> List(F,i->i[n]); [ 10,2,3 ] The computations above show that V 13 = {(0, 0, 0, 6, 1, 10), (0, 0, 1, 1, 0, 2), (1, 0, 0, 2, 0, 3)}. The package manual of numericalagps explains that, if v is a list of non-negative integers and ls is a list of lists of non-negative integers, then the function FactorizationsVectorWRTList( v, ls ) returns the set of factorizations of v in terms of the elements of ls.…”

“…In particular, a generalized numerical semigroup is a C S -semigroup such that C S = N d . Some recent results on C-semigroups are contained, for instance, in [3,15]. It is known that if S is a C S -semigroup, then S is finitely generated (so, the same occurs when S is a generalized numerical semigroup), that is, the mentioned families of submonoids of N d are classes of affine semigroups.…”