On numerical semigroups closed with respect to the action of affine maps

Ugolini, Simone

doi:10.5486/pmd.2017.7500

Cited by 2 publications

(2 citation statements)

References 2 publications

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“…In [13], numerical semigroups which are closed respect to the action of the affine maps x → αx + β, with α ∈ N \ {0} and β ∈ N, are studied. Therefore, by Corollary 8, the grepunit semigroup S a (b, n) belongs to the family studied in [13] if and only if a − (b n − 1) > 0; equivalently, a > b n − 1.…”

Section: Generalized Repunit Numerical Semigroupsmentioning

confidence: 99%

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The Frobenius problem for generalized repunit numerical semigroups

Branco¹,

Isabel²,

Ojeda³

2021

Preprint

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In this paper, we introduce and study the numerical semigroups generated by {a 1 , a 2 , . . .} ⊂ N such that a 1 is the repunit number in base b > 1 of length n > 1 and a i − a i−1 = a b i−2 , for every i ≥ 2, where a is a positive integer relatively prime with a 1 . These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a, b and n, and compute other usual invariants such as the Apéry sets, the genus or the type.

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Section: Generalized Repunit Numerical Semigroupsmentioning

confidence: 99%

“…Observe that condition a > b n − 1 corresponds to the case considered in [13] (see the comment after Corollary 8).…”

Section: The Frobenius Problemmentioning

confidence: 99%

The Frobenius problem for generalized repunit numerical semigroups

Branco¹,

Isabel²,

Ojeda³

2021

Preprint

View full text Add to dashboard Cite

show abstract

The Frobenius Problem for Generalized Repunit Numerical Semigroups

2022

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In this paper, we introduce and study the numerical semigroups generated by $$\{a_1, a_2, \ldots \} \subset {\mathbb {N}}$$ { a 1 , a 2 , … } ⊂ N such that $$a_1$$ a 1 is the repunit number in base $$b > 1$$ b > 1 of length $$n > 1$$ n > 1 and $$a_i - a_{i-1} = a\, b^{i-2},$$ a i - a i - 1 = a b i - 2 , for every $$i \ge 2$$ i ≥ 2 , where a is a positive integer relatively prime with $$a_1$$ a 1 . These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of a, b and n, and compute other usual invariants such as the Apéry sets, the genus or the type.

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On numerical semigroups closed with respect to the action of affine maps

Abstract: Abstract. In this paper we study numerical semigroups containing a given positive integer and closed with respect to the action of an affine map. For such semigroups we find a minimal set of generators, their embedding dimension, their genus and their Frobenius number.

Cited by 2 publications

References 2 publications

The Frobenius problem for generalized repunit numerical semigroups

The Frobenius problem for generalized repunit numerical semigroups

The Frobenius Problem for Generalized Repunit Numerical Semigroups

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