Higher order relations for a numerical semigroup. (English) Zbl 0818.20078

Let \(S\) be a numerical semigroup, i.e., an additive subsemigroup of \(\mathbb{N}\) with \(\mathbb{N}-S\) finite. Let \(\{b_ 0, \dots, b_ g\}\) be a minimal set of generators of \(S\). For \(J\subset \Lambda= \{0,1, \dots, g\}\), let \(b_ J= \sum_{j\in J} b_ j\), and let \(\Delta_ m= \{J\subset \Lambda\mid m-b_ J\in S\}\). Then the \(\Delta_ m\)’s form an abstract simplicial complex on the vertex set \(\Lambda\). This paper studies the square formed by the Betti numbers \(\widetilde {h}_ i (\Delta_ m)\). The properties of this square are studied for symmetric, complete intersection, and plane curve semigroups. Both combinatorial and algebraic techniques (using the semigroup algebra \(K[t^ m \mid m\in S]\)) are used.


20M14 Commutative semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
20M05 Free semigroups, generators and relations, word problems
55U05 Abstract complexes in algebraic topology
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