Modular diophantine inequalities and numerical semigroups

Rosales, J. C.; García-Sánchez, Pedro A.; Urbano-Blanco, J.M.

doi:10.2140/pjm.2005.218.379

Cited by 24 publications

(23 citation statements)

References 19 publications

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“…The following result is a consequence of Corollaries 6, 16 and 17, and Lemma 11 in [8]. Recall that a and b are positive integers, with a < b and that d = gcd{a, b} and d = gcd{a − 1, b}.…”

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confidence: 84%

“…We say that a numerical semigroup is modular if it is the set of solutions to a modular Diophantine inequality. As shown in [8], not every numerical semigroup is of this form.If S is a numerical semigroup, then the greatest integer not in S is the Frobenius number of S, denoted by g(S). The numerical semigroup S is symmetric (see [1]) if x ∈ Z \ S implies g(S) − x ∈ S (Z is the set of integers).…”

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confidence: 99%

“…Following the notation used in [8], a modular Diophantine inequality is an expression of the form ax mod b ≤ x. The set S(a, b) of integer solutions of this inequality is a numerical semigroup, that is, it is a subset of N (the set of nonnegative integers) closed under addition, 0 ∈ S(a, b) and such that N\S(a, b) has finitely many elements.…”

mentioning

confidence: 99%

“…We say that a numerical semigroup is modular if it is the set of solutions to a modular Diophantine inequality. As shown in [8], not every numerical semigroup is of this form.…”

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confidence: 99%

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Symmetric modular Diophantine inequalities

Rosales

2006

Proc. Amer. Math. Soc.

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Abstract. In this paper we study and characterize those Diophantine inequalities ax mod b ≤ x whose set of solutions is a symmetric numerical semigroup.Given two integers a and b with b = 0 we write a mod b to denote the remainder of the division of a by b. Following the notation used in [8], a modular Diophantine inequality is an expression of the form ax mod b ≤ x. The set S(a, b) of integer solutions of this inequality is a numerical semigroup, that is, it is a subset of N (the set of nonnegative integers) closed under addition, 0 ∈ S(a, b) and such that N\S(a, b) has finitely many elements. We say that a numerical semigroup is modular if it is the set of solutions to a modular Diophantine inequality. As shown in [8], not every numerical semigroup is of this form.If S is a numerical semigroup, then the greatest integer not in S is the Frobenius number of S, denoted by g(S). The numerical semigroup S is symmetric (see [1]) if x ∈ Z \ S implies g(S) − x ∈ S (Z is the set of integers). This kind of semigroup has been widely studied and characterized in the literature (see, for instance, [2,4,6]). We will say that the inequality ax mod b ≤ x is symmetric if S(a, b) is a symmetric numerical semigroup.It is well known (see, for instance, [7]) that every numerical semigroup S is finitely generated and thus there exist positive integers n 1 , . . . , n p such thatIf no proper subset of {n 1 , . . . , n p } generates S, then we say that this set is a minimal system of generators of S. Minimal systems of generators always exist and are unique (see [7]); the cardinality of a minimal system of generators of S is known as the embedding dimension of S, denoted here by e(S).Clearly, the inequality ax mod b ≤ x has the same integer solutions as the inequality (a mod b)x mod b ≤ x. Thus we may assume (and in fact we will) that a, b ∈ N and a < b. Note that S(0, b) = N is trivially symmetric.Throughout this paper (and unless otherwise stated) we will assume that a and b are positive integers, and that d = gcd{a, b} and d = gcd{a − 1, b} (gcd stands for greatest common divisor). In Proposition 4, we will show that S

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confidence: 99%

mentioning

confidence: 99%

“…We say that a numerical semigroup is modular if it is the set of solutions to a modular Diophantine inequality. As shown in [8], not every numerical semigroup is of this form.…”

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confidence: 99%

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Symmetric modular Diophantine inequalities

Rosales

2006

Proc. Amer. Math. Soc.

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“…Special cases include modular numerical semigroups [Rosales et al 2005], where I = [m/n, m/(n − 1)] (m, n ∈ ‫;)ގ‬ proportionally modular numerical semigroups [Rosales et al 2003], where I = [m/n, m/(n − s)] (m, n, s ∈ ‫;)ގ‬ and opened modular numerical semigroups [Rosales and Urbano-Blanco 2006] where I = (m/n, m/(n − 1)) (m, n ∈ ‫. )ގ‬ We consider instead arbitrary open intervals I = (a, b).…”

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confidence: 99%

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2010

Involve

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Arithmetic Varieties of Numerical Semigroups

Branco,

Ojeda,

Rosales

2024

Results Math

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In this paper we present the notion of arithmetic variety for numerical semigroups. We study various aspects related to these varieties such as the smallest arithmetic that contains a set of numerical semigroups and we exhibit the rooted tree associated with an arithmetic variety. This tree is not locally finite; however, if the Frobenius number is fixed, the tree has finitely many nodes and algorithms can be developed. All algorithms provided in this article include their (non-debugged) implementation in GAP.

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Modular diophantine inequalities and numerical semigroups

Cited by 24 publications

References 19 publications

Symmetric modular Diophantine inequalities

Symmetric modular Diophantine inequalities

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Arithmetic Varieties of Numerical Semigroups

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