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A geometrical characterization of proportionally modular affine semigroups. (English) Zbl 1521.20129

Summary: A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality \(f_1 x_1 + \cdots + f_n x_n \bmod b \le g_1 x_1 + \cdots + g_n x_n\), where \(g_1, \dots ,g_n, f_1,\ldots ,f_n \in \mathbb{Z}\), and \(b \in \mathbb{N}\). In this work, a geometrical characterization of these semigroups is given. On the basis of this geometrical approach, some algorithms are provided to check if a semigroup \(S\) in \(\mathbb{N}^n\), with \(\mathbb{N}^n{\setminus} S\) a finite set, is a proportionally modular affine semigroup.

MSC:

20M14 Commutative semigroups
20-04 Software, source code, etc. for problems pertaining to group theory
20-08 Computational methods for problems pertaining to group theory
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References:

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