Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains

Barucci, Valentina; Dobbs, David E.; Fontana, Marco

doi:10.1090/memo/0598

Cited by 209 publications

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“…Some of the characterizations that we give here coincide with those given in [2]. We first present some notations.…”

Section: Conditions Equivalent To the Arf Conditionmentioning

confidence: 66%

“…For each k ≥ 1 , we consider the set kM In what follows, we will give some results about Lipman semigroups that can be found in [2]. …”

Section: Lipman Sequence Of a Numerical Semigroupmentioning

confidence: 99%

“…A subset S ⊆ N 0 satisfying (i) 0 ∈ S (ii) x, y ∈ S ⇒ x + y ∈ S (iii) |N 0 \ S| < ∞ is called a numerical semigroup. It is well known (see, for instance, [2,3,5]) that the condition (iii) above is equivalent to saying that the greatest common divisor gcd(S) of elements of S is 1 .…”

Section: Introductionmentioning

confidence: 99%

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Arf numerical semigroups

İlhan¹,

Karataş²

2017

Turk J Math

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The aim of this work is to exhibit the relationship between the Arf closure of a numerical semigroup S and its Lipman semigroup L(S). This relationship is then used to give direct proofs of some characterizations of Arf numerical semigroups through their Lipman sequences of semigroups. We also give an algorithmic construction of the Arf closure of a numerical semigroup via its Lipman sequence of semigroups.

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“…Some of the characterizations that we give here coincide with those given in [2]. We first present some notations.…”

Section: Conditions Equivalent To the Arf Conditionmentioning

confidence: 66%

“…For each k ≥ 1 , we consider the set kM In what follows, we will give some results about Lipman semigroups that can be found in [2]. …”

Section: Lipman Sequence Of a Numerical Semigroupmentioning

confidence: 99%

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Arf numerical semigroups

İlhan¹,

Karataş²

2017

Turk J Math

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“…We denote by PF(S) the set of pseudo-Frobenius numbers of S. From the definition it follows that F(S) = max{PF(S)}. The cardinal of PF(S) is an important invariant of S which is called the type of S (see [1]). In [3] …”

Section: Theorem 22 Under the Above Conditions We Havementioning

confidence: 99%

The Frobenius problem for numerical semigroups with embedding dimension equal to three

Robles-Pérez

Rosales

2012

Math. Comp.

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“…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.…”

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confidence: 99%

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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Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains

Cited by 209 publications

References 0 publications

Arf numerical semigroups

Arf numerical semigroups

The Frobenius problem for numerical semigroups with embedding dimension equal to three

Symmetric modular Diophantine inequalities

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