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Ramírez Alfonsín, Jorge L.; Rødseth, Øystein J.

Numerical semigroups: Apéry sets and Hilbert series. (English) Zbl 1200.20047

Semigroup Forum 79, No. 2, 323-340 (2009).
A numerical AA-semigroup \(S\) is defined as the semigroup generated by positive integers \(a,a+d,a+2d,\dots,a+kd,c\), assumed to be relatively prime. If \(g\) is the largest integer not in \(S\), then \(S\) is called symmetric if \(S\cup(S-g)\) equals the ring of integers.
The authors characterize symmetric AA-semigroups, and give an algorithm which permits the use of a formula due to the second author [J. Reine Angew. Math. 307/308, 431-440 (1979; Zbl 0395.10021)] to determine the Apéry set of \(S\) [R. Apéry, C. R. Acad. Sci., Paris 222, 1198-1200 (1946; Zbl 0061.35404)], hence also the corresponding Frobenius number.
Reviewer: Władysław Narkiewicz (Wrocław)

Cited in 19 Documents

MSC:

20M13 Arithmetic theory of semigroups
11D07 The Frobenius problem
20M14 Commutative semigroups
11D04 Linear Diophantine equations

Keywords:

almost arithmetic semigroups; AA-semigroups; Frobenius numbers; Apéry sets; Hilbert series; almost arithmetic progressions

Citations:

Zbl 0395.10021; Zbl 0061.35404
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Full Text: DOI

References:

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