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The integral closure of a finitely generated monoid and the Frobenius problem in higher dimensions. (English) Zbl 0818.20079

Bonzini, C. (ed.) et al., Semigroups: algebraic theory and applications to formal languages and codes. Proceedings of the international conference on semigroups, held in Luino, Italy, from 22nd to 27th June, 1992. Singapore: World Scientific. 86-93 (1993).
Let \(V\) be a finite-dimensional vector space over the field \(\mathbb{Q}\) of rational numbers. A subset \(C\subseteq V\) is called a polyhedral cone if there exists a homomorphism \(\varphi: \mathbb{Q}^ s\to V\) for some \(s\geq 1\) such that \(C= \varphi (\mathbb{Q}^ s_{\geq 0})\). It is proved that for a polyhedral cone \(C\subseteq V\) and a finitely generated subgroup \(\Gamma\subseteq V\) the set \(C\cap \Gamma\) is a finitely generated monoid. For a finitely generated submonoid \(H \subseteq V\), \(\text{cone} (C)= \text{cone} (H)\) and for the group \(\Gamma\) generated by \(H\) in \(V\) there exists some \(v\in H\) such that \(v+ (C\cap \Gamma) \subseteq H\). From this result the author deduces a corollary on solvability and the number of solutions of a system of linear diophantine equations. If \(\Gamma\) is an abelian group and \(H \subseteq \Gamma\) a submonoid then the monoid \(\{x\in \Gamma\mid nx\in H\) for some natural number \(n\}\) is called the integral closure of \(H\) in \(\Gamma\). If \(\Gamma\) is a finitely generated abelian group and \(H \subseteq \Gamma\) a finitely generated submonoid then the integral closure of \(H\) in \(\Gamma\) is also a finitely generated monoid. In conclusion the notion of Krull monoid is considered as special case of integrally closed monoid.
For the entire collection see [Zbl 0799.00023].

MSC:

20M14 Commutative semigroups
15A03 Vector spaces, linear dependence, rank, lineability
11D04 Linear Diophantine equations
13B22 Integral closure of commutative rings and ideals
20K99 Abelian groups
20M05 Free semigroups, generators and relations, word problems
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