Normality and Covering Properties of Affine Semigroups

S̄ = {x ∈ gp(S) | mx ∈ S for some m > 0}. One calls S normal if S = S̄. For simplicity we will often assume that gp(S) = Z; this is harmless because we can replace Z by gp(S) if necessary. The rank of S is the rank of gp(S). We will only be interested in the case in which S ∩ (−S) = 0; such affine semigroups will be called positive. The positivity of S is equivalent to the pointedness of the cone C(S) = R+S generated by S in R; one has rankS = dimC(S). (All the cones appearing in this paper have their apex at the origin.) The normality of S can now be characterized geometrically: one obviously has S̄ = C(S) ∩ Z, and so S is normal if and only if S = C(S) ∩ Z. Conversely, every convex, pointed, finitely generated rational cone C ⊂ R yields a normal affine semigroup S(C) = C ∩ Z (this semigroup is finitely generated by Gordan’s lemma). If we simply speak of a cone C in the following, then it is always assumed that C is convex, pointed, finitely generated, and rational. Since dimC(S) = rankS, we often call the dimension of a cone its rank. Each positive affine semigroup S can be embedded into Z+ , m = rankS (one uses m linearly independent integral linear forms representing support hyperplanes of C(S)). Therefore every element can be written as a sum of irreducible elements, and the set of irreducible elements of S is finite since S is finitely generated. For a cone C the set of irreducible elements of S(C) is often called the Hilbert basis Hilb(C) of C in the combinatorial literature, a convention we will follow. More generally, we also call the set of irreducible elements of a positive affine semigroup S its Hilbert basis and denote it by Hilb(S). This paper is devoted to a discussion of sufficient and potentially necessary conditions for the normality of S in terms of combinatorial properties of Hilb(S). As far as the necessity is concerned, these conditions can of course be formulated in terms of Hilbert bases of rational cones. The first property under consideration is the existence of a unimodular (Hilbert) cover: