Characterizing affine $${\mathcal {C}}$$-semigroups

Abstract

Let $${\mathcal {C}} \subset {\mathbb {N}}^p$$ C ⊂ N p be a finitely generated integer cone and $$S\subset {\mathcal {C}}$$ S ⊂ C be an affine semigroup such that the real cones generated by $${\mathcal {C}}$$ C and by S are equal. The semigroup S is called $${\mathcal {C}}$$ C -semigroup if $${\mathcal {C}}\setminus S$$ C \ S is a finite set. In this paper, we characterize the $${\mathcal {C}}$$ C -semigroups from their minimal generating sets, and we give an algorithm to check if S is a $${\mathcal {C}}$$ C -semigroup and to compute its set of gaps. We also study the embedding dimension of $${\mathcal {C}}$$ C -semigroups obtaining a lower bound for it, and introduce some families of $${\mathcal {C}}$$ C -semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a $${\mathcal {C}}$$ C -semigroup into irreducible $${\mathcal {C}}$$ C -semigroups.

Conductors of Abhyankar-Moh semigroups of even degrees

“…In this section, we provide an algorithm to compute F p (S) for any p ∈ N. Recall that to solve the problem for p = 0, you can use the results appearing in [7]. The following result is the key to obtain such an algorithm for p ≥ 1.…”

“…Moreover, we give two improved algorithms for the cases p = 1 and p = 2. The case p = 0 was solved in [7]. In that paper, the authors characterize the affine semigroups S such that C(S) \ S is finite, and an algorithm to compute its gap sets is introduced.…”

On Some Numerical Semigroup Transforms

“…Let S and C be affine semigroups, S ⊆ C. We recall that for C = C S a characterization so that |C S \ S| < ∞ has been provided in [10,Theorem 9] and in the same paper the authors provide a procedure to compute C S \ S.…”

“…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.…”

“…A natural problem is to consider a set A ⊆ N d and characterize when the set S = A is a C S -semigroup. A result of this type has been provided firstly in [6,Theorem 2.8] for generalized numerical semigroups and later in [10,Theorem 9] for the general case of C Ssemigroups. Having in mind these results, in this paper we consider a further generalization for the property of cofiniteness of an affine semigroup S ⊆ N d , considering the case S is cofinite in another affine semigroup C ⊆ N d .…”