INdAM meeting: International meeting on numerical semigroups - Cortona 2018

Let $\mathbb{K}$ be a field and let $f_1,\ldots,f_s$ be nonzero elements of the ring of formal power series $\mathbf{R}=\mathbb{K}[\![x_1,\ldots,x_n]\!]$. Let $\mathbf{A}=\mathbb{K}[\![f_1,\ldots,f_s]\!]$ be the $\mathbb{K}$-subalgebra of $\mathbf{R}$ generated by $\lbrace f_1,\ldots,f_s\rbrace$. Suppose that $l(\mathbf{R}/\mathbf{A})$ is finite, and let $L$ be a linear form on $\mathbb{R}^n$ with nonnegative coefficients, then $L$ defines a filtration on $\mathbf{R}$. Any nonzero element $f$ of $\mathbf{R}$ has a decomposition into a sum of $ L$-homogeneous components. The initial form of $f$, denoted in$(f,L)$, is the component of $f$ of smallest degree with respect to $L$.The greatest exponent of $\operatorname{in}(f,L)$ with respect to a well ordering on $\mathbb{N}^n$, denoted exp$(f,L)$, is called the $L$-exponent of $f$, and the set of $L$-exponents of elements of $\mathbf{A}$ is an affine semigroup. We denote it by $\operatorname{exp}(\mathbf{A},L)$.

In this talk we consider the problem of the stability of exp$(\mathbf{A},L)$ when the coefficients of $L$ vary in $\mathbb{R}$. We first prove that the set $E$ of exp$(\mathbf{A},L)$ is finite. Next we prove that there exists a fan in $\mathbf{R_+}^n$ (viewed as the set of linear forms) and a correspondance between the elements of $E$ and the set of cones of this fan: Given a cone $C$, exp$(\mathbf{A},L)$ does not depend on $L\in C$ and it is an elment of $E$.

For a nonnegative integer $g$, we denote the set of numerical semigroups of genus $g$ by $\mathcal{S}_g$ and its cardinality by $n_g$. Zhai proved that $\frac{n_{g+1}}{n_g}$ approaches the golden ratio, hence $n_g < n_{g+1}$, for large enough $g$. It remains as an open problem to decide if $n_g < n_{g+1}$ holds true for all $g$.

For nonnegative integers $g$ and $\gamma$, we denote the set of numerical semigroups of genus $g$ and $\gamma$ even gaps by $\mathcal{S}_{\gamma}(g)$ and its cardinality by $N_{\gamma}(g)$. Torres proved that if $S \in \mathcal{S}_{\gamma}(g)$, then $2g \geq 3\gamma$; hence $n_g = \sum_{\gamma = 0}^{\lfloor 2g/3 \rfloor} N_\gamma(g)$. B. and Torres studied $N_{\gamma}(g)$; it was proved that $N_{\gamma}(g) = N_{\gamma}(3\gamma)$, if $g \geq 3\gamma$ and, otherwise, $N_{\gamma}(g) < N_{\gamma}(3\gamma)$. From the surjective map $\mathbf{x}: \mathcal{S}_{\gamma}(g) \to \mathcal{S}_\gamma, S \mapsto S/2$, $N_{\gamma}(g)$ can be computed as $\sum_{T \in \mathcal{S}_\gamma} \#\mathbf{x}^{-1}(T)$.

In this talk, we study the problem of computing the numbers $N_\gamma(g)$ by using the multiplicity of $T \in \mathcal{S}_\gamma$ and its Apéry set. This leads to a polytope problem and has a close relation with the problem of deciding if the sequence $(n_g)$ is increasing.

We provide a characterization of the Arf property in both the numerical duplication of a numerical semigroup and in a member of a family of quotients of the Rees algebra studied in V. Barucci, M. D'Anna, and F. Strazzanti. "A family of quotients of the Rees algebra." Communications in Algebra 43.1 (2015): 130-142.

Consider a sequence of AG codes evaluating at a set of evaluation points $P_1,\ldots,P_n$ the functions having only poles at a defining point $Q$, with the sequence of codes satisfying the isometry-dual condition (i.e. containing at the same time primal and their dual codes). We prove a necessary condition under which, after taking out a number of evaluation points (i.e. puncturing), the resulting AG codes can still satisfy the isometry-dual property. The condition has to do with the so-called maximum sparse ideals of the Weierstrass semigroup of $Q$.

Let $\Gamma$ be a numerical semigroup. The Leamer monoid $S_\Gamma^s$, for $s\in \mathbb{N}\backslash \Gamma$, is the monoid consisting of arithmetic sequences of step size $s$ contained in $\Gamma$. In this note, we give a formula for the $\omega$-primality of elements in $S_\Gamma^s$ when $\Gamma$ is an numerical semigroup generated by a arithmetic sequence of positive integers.

We say that an element of an additive commutative cancellative monoid is a molecule if it has a unique factorization, i.e., if it can be expressed in a unique way as a sum of irreducibles. In this paper, we study the sets of molecules of Puiseux monoids (additive submonoids of $Q \ge 0$). In particular, we present results on the possible cardinalities of the sets of molecules and the sets of those molecules failing to be irreducibles for numerical semigroups and general Puiseux monoids. We also construct infinitely many non-isomorphic Puiseux monoids all whose molecules are irreducible. Finally, we target Puiseux monoids generated by rationals with single-prime denominators, and provide a characterization for their sets of molecules.

In a paper published 40 years ago (1978), Wilf proposed two problems on numerical semigroups, which are combinatorial in nature. One of the problems is about counting numerical semigroups (how does the number of numerical semigroups with a given conductor behaves when the conductor grows?). Nice and very satisfactory results related to the problem of counting were obtained (although most of them considering counting by genus). For instance, the number of numerical semigroups of a given genus behaves, when the genus grows, like the Fibonacci sequence. The other problem proposed, nowadays known as Wilf's conjecture, relates some of the most studied combinatorial invariants in the area. To the best of my knowledge, although much progress has been made, the problem is still unsolved. As frequently happens with problems that draw the attention of many researchers and last for long, independently of being solved, they constitute important sources of other interesting problems. Our aim is to talk about these problems proposed by Wilf and other problems originated by them, by mentioning some of the results known as well as refer some recent progress made.

This is joint work with Jean Fromentin.

For any $g \ge 0$, let $n_g$ denote the number of numerical semigroups of genus $g$. Ten years ago, Maria Bras-Amorós made some remarkable conjectures revealing a Fibonacci-like behavior of the numbers $n_g$. In this talk, we propose a new point of view on numerical semigroups which allows us to make some partial progress towards these conjectures.

I will talk about related topics about RF-matrices (row-factorization matrices) of almost symmetric semigroups of embedding dimension four.

We present an effective method to compute Weierstrass semigroups for plane curves over finite fields, with the aid of Hamburger-Noether expansions and adjoints. This algorithm is implemented in the computer algebra system Singular. If the point is at infinity, we also present an alternative method by using the Abhyankar-Moh algorithm with some triangulation procedure. This problem is related to the effective construction of Algebraic Geometry error-correcting codes.

We consider symmetric (not complete intersection) numerical semigroups generated by four, five and six elements, and derive inequalities for degrees of syzygies of such semigroups and find the lower bounds $F$ for their Frobenius numbers. We study a special case of such semigroups, which satisfy the Watanabe Lemma, and show that the lower bound $W$ for the Frobenius numbers of such semigroup is stronger than $F$.

Bibliography

Semigroup Forum, 93 (2016), 423-426; Ramanujan Journal, 43 (2017), 465-491; INTEGERS: The Electronic J. of Comb. Number Theory, 18 (2018), # A44.

Joint work with Maria Bras-Amorós.

For a numerical semigroup, we consider the set of generators which are larger than its Frobenius number and show how to produce in a fast way the corresponding sets for its children in the semigroup tree. This allows us to present an efficient algorithm for computing the descendants of a numerical semigroup up to a given genus.

John Milnor proved in his celebrated book [M] the formula

(M)$ \mu=2\delta-r+1,$

where $\mu$ is the Milnor number $\mu$, $\delta$ the double point number and $r$ the number of branches of a plane curve singularity. The Milnor’s proof of (M) is based on topological considerations. A proof given by Risler [R] is algebraic and shows that (M) holds in characteristic zero.

On the other hand Melle and Wall based on a resultd by Deligne [D] proved the inequality $\mu\geq 2\delta-r+1$ in arbitrary characteristic and showed that the Milnor formula holds if and only if the singularity has not wild vanishing cycles [MH-W]. In the sequel we will call a tame singularity any plane curve singularity verifying (M).

Recently some papers on the singularities satisfying (M) in characteristic $p$ appeared. In [B-G-M] the authors showed that planar Newton non-degenerate singularities are tame. Different notions of non-degeneracy for plane curve singularities are discussed in [G-N]. In [N] the author proved that if the characteristic $p$ is greater than the kappa invariant then the singularity is tame. In [GB-P1] and [H-R-S] the case of irreducible singularities is investigated.

In the irreducible case the study is given using the semigroup associated with the singularity. We will say that this semigroup is tame if and only if the singularity of the branch is tame.

Our aim is to give a sharpened version of the main result of [GB-P1].

This talk is based on the results of [GB-P2].

Bibliography

[B-G-M] Boubakri, Y., Greuel G-M, Markwig, T.: Invariants of hypersurface singularities in positive characteristic. Rev. Mat. Complut. 25, 61-85, (2010)

[D] Deligne, P. La formule de Milnor, Sem. Geom. algébrique, Bois-Marie 1967-1969, SGA 7 II, Lect. Notes Math., 340, Exposé XVI, 197-211 (1973).

[GB-P1] García Barroso, E.; Płoski, A.: The Milnor number of plane irreducible singularities in positive characteristic. Bulletin of the London Mathematical Society 48(1), 94-98 (2016). doi: 10.1112/blms/bdv095

[GB-P2] García Barroso, E.; Płoski, A.: On the Milnor formula in arbitrary characteristic. Accepted for publication in Singularities, Algebraic Geometry, Commutative Algebra and Related Topics. Festschrift for Antonio Campillo on the Occasion of his 65th Birthday. G.-M. Greuel, L. Narvaéz and S. Xambó-Descamps eds. Springer, 2018.

[G-N] Greuel, G-M, Nguyen, H.D.: Some remarks on the planar Kouchnirenko’s theorem. Rev. Mat. Complut 25, 557-579 (2012)

[H-R-S] Hefez, A, Rodrigues, J.H.O., Salomão, R.: The Milnor number of a hypersurface singularity in arbitrary characteristic. arXiv:1507.03179v1 (2015)

[MH-W] Melle-Hernández, A., Wall C.T.C.: Pencils of cuves on smooth surfaces. Proc. Lond. Math. Soc., 83(2), 257-278 (2001).

[M] Milnor, J. W.: Singular points of complex hypersurfaces Princeton University Press (1968)

[N] Nguyen, H.D.: Invariants of plane curve singularities, and Plücker formulas in positive characteristic. Annales Inst. Fourier 66, 2047-2066 (2016)

[R] Risler, J.J.: Sur l’idéal jacobien d’une courbe plane. Bull. Soc. Math. Fr., 99(4), 305-311 (1971)

This is a joint work with Kolja Knauer from Aix-Marseille Université.

Let $P$ be a partially ordered set with a global minimum 0. In the chomp game on $P$, two players alternatively pick an element of $P$. Whoever is forced to pick 0 loses the game. A move consists of picking an element $x$ in $P$ and removing its upset, i.e., all the elements that are larger or equal to $x$. The type of questions one wants to answer is: for a given $P$, has either of the players a winning strategy? and, in that case, can a strategy be devised explicitly. Many combinatorial games such as NIM, DIVISORS or the chocolate-bar-game are instances of chomp. In this talk we consider P to be a numerical semigroup and the order: $x \leq y$ if and only $y - x \in P$. We will show how several algebraic properties of the semigroup can be translated in terms of winning strategies. We will also prove that even if $P$ is infinite, the problem of determining which player has a winning strategy is decidable. Finally, we will briefly discuss some generalizations of our results for more general posets with a semigroup structure.

A commutative and cancellative monoid $H$ is said to be primary if for each two non-invertible elements $a, b \in H$ there is an $n \in \mathbb{N}$ such that $a \mid b^n$. Clearly, numerical monoids are primary and a commutative integral domain is primary if and only if it is one-dimensional and local. Let $H$ be a primary monoid. If an element $a \in H$ has a factorization $a=u_1 \cdot \ldots \cdot u_k$, where $u_1, \ldots, u_k$ are irreducible elements, then $k$ is called a factorization length of $a$. The set $\mathsf L (a) \subset \mathbb N$ of all possible factorization lengths of $a$ is called the set of lengths of $a$. We give an overview of recent results on the system $\mathcal{L}(H) = \{ \mathsf{L}(a) \mid a \in H\}$ of sets of lengths of $H$. Among others, we present the result that for every finite nonempty subset $L \subset \mathbb{N}_{\ge 2}$ there are a numerical monoid $H$ and a squarefree element $a \in H$ whose set of lengths $\mathsf{L}(a)$ equals $L$ [Ge-Sc18e].

References

[Ge-Sc18e] A. Geroldinger and W.A. Schmid, A realization theorem for sets of lengths in numerical monoids, Forum Math., to appear.

This is a joint work with Hema Srinivasan (Missouri University, USA).

We construct the minimal free resolution of the semigroup ring $k[C]$ in terms of the minimal resolutions of $k[A]$ and $k[B]$ when $\langle C\rangle$ is a semigroup obtained by gluing two semigroups $\langle A\rangle$ and $\langle B\rangle$. Using our explicit construction, we compute the Betti numbers and Hilbert series. We also show that if $k[A]$ and $k[B]$ have a differential graded algebra structure, then $k[C]$ also has such a structure and we explicitly give the multiplication on the resolution of $k[C]$ in terms of those of $k[A]$ and $k[B]$.

Here we consider a rational generalization of a numerical semigroup algebra by allowing the exponents of the generating monomials to be positive rationals instead of naturals. We construct three families of these generalized semigroups algebras. The first family consists of semigroup algebras of certain atomic Puiseux monoids called primary. Properties of primary Puiseux monoids will allow us to show that the semigroup algebras they determine are atomic. The second family of algebras is constructed over a given algebraically closed field by using integrally closed Puiseux monoids. As we shall prove, the algebraically closedness of the field and the integrally closedness of the monoids ensure that the semigroup algebras they determine are antimatter Bezout domains. Then, we use a class of divisible Puiseux monoids to construct our last family of antimatter semigroup algebras over any perfect field of finite characteristic.

This is a joint work in progress with Marco D'Anna and Vincenzo Micale.

Good semigroups (in $\mathbb{N}^2$) arise as value semigroups of curve singularities with two branches, but the class of good semigroups is larger than the class of value semigroups. Good semigroups are a natural generalization of numerical semigroups, but one main difference is that they are not finitely generated. This fact implies many problems in the study of good semigroups: for example, while there is a natural generalization of symmetric numerical semigroups, no definition of complete intersection good semigroups has been given. In this talk I will present some new results about the Apéry set of a good semigroup, both in the general and in the symmetric case, showing how to generalize many properties of the Apéry set of a numerical semigroup. In particular, these results could lead to a definition of complete intersection good semigroup (in $\mathbb{N}^2$).

This is a joint work with Raheleh Jafari.

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The project is to study whether a flat rectangular algebra is complete intersection. Along this direction, special types of algebras generated by few monomials are work out in detail.

In this work we give a new characterization of Arf numerical semigroups in terms of the Apéry set and use it to parametrize Arf numerical semigroups with multiplicity 9 and 10.

We investigate the Gorenstein property of endomorphism rings on curve singularities in terms of the semigroup of values. In particular, we characterize the Gorenstein algebroid curves for which the endomorphism ring of the maximal ideal is Gorenstein.

Joint work with P. A. García-Sánchez and U. Krause.

A reduced, cancelative and commutative monoid $M$ is called an inside factorial monoid with base $Q\subset M$, if for every $x\in M$ there exists a positive integer $n(x)$ such that $n(x)x=\sum_{q\in Q} x(q)q$ with $x(q)$ unique and $x(q)=0$ except for finitely many $q\in Q$. We observe that the set of nonnegative integer solutions of the equation $a_1x_1+\cdots +a_{r-1}x_{r-1}=a_rx_r$, with $a_1,\ldots ,a_r$ positive integers is a inside factorial monoid. We use this idea to give some results for this set.

Joint work with P. A. García-Sánchez and A. M. Robles-Pérez.

We show that the number of numerical semigroups with multiplicity three, four or five and fixed genus is increasing as a function in the genus. To this end we use the Kunz polytope for these multiplicities. Counting numerical semigroups with fixed multiplicity and genus is then an integer partition problem with some extra conditions (those of membership to the Kunz polytope). For the particular case of multiplicity four, we are able to prove that the number of numerical semigroups with multiplicity four and genus $g$ is the number of partitions $x+y+z=g+6$ with $0<x\le y\le z$, $x\neq 1$, $y\neq 2$ and $z\neq 3$.

Joint work with J. I. García-García, D. Marín-Aragón, J. C. Rosales, and A. Vigneron-Tenorio

Given $m\in\mathbb{N}$, a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $[m,2m-1]$. In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf’s conjecture is checked in the tree associated to some packed numerical semigroups

In this talk, I will show you when a numerical semigroup ring obtained by gluing is a generalized Gorenstein local (for short, GGL) ring, which is a new class of local rings introduced by Shiro Goto and Shinya Kumashiro recently. This talk will be divided into two parts:

(1) The definition and basic facts on GGL numerical semigroup rings,

(2) The main theorem and sketch of proof.

This is a joint work together with W. Tenòrio and F. Torres.

We investigate the structure of the generalized Weierstraß semigroups at several points on a curve defined over a finite field. We present a description of these semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch spaces. This characterization allows us to show that the Poincaré series associated with generalized Weierstraß semigroups carry essential information to describe entirely their respective semigroups.

Joint work with M. B. Branco and J. C. Rosales.

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP package NumericalSgps.

Joint work with José Carlos Rosales.

We are interested in computing the Frobenius number of numerical semigroups that are generated by sequences of binomial coefficients. In addition, some properties of such numerical semigroups are shown.

This work is supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía, Industria y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343.

We talk about evaluation codes defined on abstract toric varieties, whose lengths and dimensions can be computed via multigraded Hilbert functions of the corresponding multigraded lattice ideals. One can confirm when these ideals or equivalently corresponding pointed semigroups are complete intersection, using mixed dominating matrices with columns constituting a basis for the lattice in question.

We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.

Star operations are closure operations that are classically defined on the set of fractional ideals of an integral domain; their definition can be extended in a natural way to cancellative semigroups. In this talk, we study the set $\mathrm{Star}(S)$ of star operations on a numerical semigroup $S$; more precisely, we are interested in bounding and, if possible, in determining its cardinality. We also consider the set $\mathcal{G}_0(S)$ of (classes of) non-divisorial ideals of $S$, which can be given a natural partial order induced from $\mathrm{Star}(S)$, and study how the properties of $S$ are reflected in the shape of $\mathcal{G}_0(S)$. Finally, we show the relationship between these results and the ring case.

This is joint work with Juergen Herzog.

We present new criteria for deciding the Cohen-Macaulay property for projective monomial curves using Groebner bases.

Joint work with Marco D’Anna and Raheleh Jafari.

Let $S$ be a numerical semigroup, $E$ be a proper ideal of $S$ and $b \in S$ be an odd integer. The numerical duplication $S \! \Join^b \! E$ of $S$ with respect to $E$ and $b$ is a numerical semigroup which arises in commutative algebra. This construction has had interesting applications both in numerical semigroup theory and commutative algebra; for instance, the study of its associated graded ring led to construct the first known examples of symmetric numerical semigroups whose Hilbert function is decreasing (in some level). In this talk we are interested to deepen the knowledge of the associated graded ring of $S \! \Join^b \! E$, in particular studying its Cohen-Macaulayness, Gorensteinness, and other similar properties.

Joint work with A. Oneto

Let $R=k[\![S]\!]$ where $S$ is a numerical semigroup minimally generated by four elements. We prove that the Hilbert function $H_R$ of $R$ is non decreasing under certain conditions on the Apéry set of $S$. In particular, $H_R$ has this property when $S$ has multiplicity $\leq 13$.

This is a joint work with Dimitra Kosta.

A monomial curve $C$ in $A^3$ has Markov complexity $m(C)$ two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there in no $d\in N$ such that $m(C)\leq d$ for all monomial curves $C$ in $A^4$. The same result is true even if we restrict to complete intersections. We extend this result for all monomial curves in $A^m$, where $m\geq 4$.

We will discuss the robustness property of monomial curves.

Joint work with J. Ossorio

While quantum computing is still far from being a daily-life reality (?) quantum computation is making quick progress. In particular, interesting questions arise when one tries to design quantum algorithms to solve established hard problems. In this talk we will address the Frobenius Diophantine Problem from a quantum computation viewpoint.

It is well-known that Gorenstein rings have symmetric Hilbert series. For plane curves it is possible to give a definition of Poincaré series, which encodes important topological informations of the curve. In this talk we extend the definition to the class of good semigroup ideals, which contains value semigroups of algebraic curves. We show that under suitable assumptions, for a good semigroup $E$ and canonical ideal $K$, the Poincaré series of $K-E$ is symmetric to the Poincaré series of $E$.

The main result is an asymptotic formula for the number of symmetric semigroups with 3 generators where the biggest generator is a fixed large number. The proof is based on analytic methods.

Joint work with J. I. García-García and D. Marín-Aragón.

Let $\mathcal{C}\subset \mathbb{Q}^p$ be a rational cone. An affine semigroup $S\subset \mathcal{C}$ is a $\mathcal{C}$-semigroup whenever $(\mathcal{C}\setminus S)\cap \mathbb{N}^p$ has only a finite number of elements. A tree of $\mathcal{C}$-semigroups is studied given a method to generate it. Besides, the $\mathcal{C}$-semigroups with minimal embedding dimension are studied. Moreover, Wilf's conjecture for numerical semigroups is extend to affine $\mathcal{C}$-semigroups and some families of $\mathcal{C}$-semigroups fulfilling this extended conjecture are given.

References

[1] J. I. García-García, D. Marín-Aragón, A. Vigneron-Tenorio, An extension of Wilf's conjecture to affine semigroups, Semigroup Forum 96 (2018), 396–408.

Let $H = \langle n_1,n_2, n_3, n_4\rangle$ be a symmetric numerical semigroup generated by 4 elements. Bresinski proved that the defining ideal of the semigroup ring $k[H]$ is generated by either 3 or 5 elements. But his proof of this theorem is rather difficult and complicated. We will give a shorter proof using the minimal free resolution of $k[H]$ over a polynomial ring and Buchsbaum-Eisenbud structure theorem of codimension 3 Gorenstein rings.

This is joint work with Craig Huneke and Srikanth Iyengar.

Let $R$ be a Gorenstein local integral domain of dimension one. An ideal $I$ of $R$ is said to be rigid provided every short exact sequence beginning and ending with $I$ splits; equivalently, the tensor product of $I$ with its inverse is torsion-free. A conjecture, still unresolved, is that every rigid ideal is principal. We give a proof of the conjecture if the multiplicity of $R$ is at most 8, or at most 10 if $R$ is a complete intersection. Also, the conjecture holds for rings of minimal multiplicity. Numerical semigroup rings seem to be the place to look for counterexamples. For one thing, the only examples of non-principal ideals with torsion-free tensor product occur over numerical semigroup rings.

In this talk we explore the properties of the Arf good subsemigroups of $\mathbb{N}^n$, with $n\geq2$. We give a way to compute all the Arf semigroups with a given collection of multiplicity branches. We also deal with the problem of determining the Arf closure of a set of vectors and of a good semigroup, extending the concept of characters of an Arf numerical semigroup to Arf good semigroups. Furthermore we present some procedures to calculate the set of the Arf good semigroups with a given conductor and with a given genus.