ARF numerical semigroups








M. B. Branco
Universidade de Évora, Portugal

J. C. Rosales, P. A. García-Sánchez, J. I. García-García
Universidad de Granada, Spain








A numerical semigroup $S$ is a subset of ${\mathbb{N}}$ that is closed under addition, $0\in S$ and generates ${\mathbb{Z}}$ as a group ( ${\mathbb{Z}}$ and ${\mathbb{N}}$ denote the set of integers and nonnegative integers, respectively). It is well known (see for instance [2,8,13]) that if $S$ is a numerical semigroup, then the set ${\mathbb{N}}\setminus S$ has finitely many elements. The greatest integer not belonging to $S$ is called the Frobenius number of $S$, usually denoted by ${\rm g}(S)$. Moreover, $S$ admits a unique minimal system of generators $\{n_1<\cdots <n_p\}$ (that is, $S=\{\sum_{i=1}^p a_i n_i~\vert~ a_1,\ldots,a_p\in {\mathbb{N}}\}$ and no proper subset of $\{n_1,\ldots,n_p\}$ generates $S$). The integers $n_1$ and $p$ are known as the multiplicity and embedding dimension of $S$, and they are denoted by ${\rm m}(S)$ and $\mu(S)$, respectively. For a given $A\subseteq {\mathbb{N}}$, we will denote by $\langle A\rangle$ the submonoid of ${\mathbb{N}}$ generated by $A$. The monoid $\langle A\rangle$ is a numerical semigroup if and only if $\gcd(A)=1$ ($\gcd$ stands for greatest common divisor). There is a large amount of literature concerning the study of one-dimensional analitically unramified domains via their valuation semigroups (see for instance [3,5,6,7,10,15,16]). One of the properties studied for this kind of rings using this approach has been the Arf property. From [1], Lipman in [11] introduces and motivates the study of Arf rings; the characterization given in that paper of Arf rings in terms of its semigroup of values gives rise to the notion of Arf semigroup (see also [2] for the connection between the Arf property of a one-dimensional analitically irreducible domain and the Arf property of its semigroup of values). In [14,4] it is studied the relationship between the Pythagorean property of a real curve germ and the Arf property of its value numerical semigroup. A numerical semigroup $S$ is an Arf numerical semigroup if for every $x,y,z\in S$ such that $x\geq y\geq z$, we have that $x+y-z\in S$ (see [2, Theorem I.3.4] for fifteen alternative characterizations of this property). We deduce that the intersection of two Arf numerical semigroups is again an Arf numerical semigroup. This allows us to define the Arf numerical semigroup generated by $A$ ($\gcd(A)=1$) as the intersection of all Arf numerical semigroups containing $A$ (and thus $\langle A\rangle$), and will denote it by ${\rm Arf}(A)$. If $S={\rm Arf}(A)$, we say that $A$ is an Arf system of generators of $S$, and we will say that $A$ is minimal if no proper subset of $A$ is an Arf system of generators of $S$. For a numerical semigroup $S$, ${\rm Arf}(S)$ will be also called the Arf closure of $S$. Observe that if we are given $A\subseteq {\mathbb{N}}$ with $\gcd(A)=1$, then ${\rm Arf}(A)$ must contain the set of all the elements of the form $x+y-z$ with $x,y,z\in \langle A\rangle$ and $x\geq y\geq z$. It must also contain the set of elements that are derived from those obtained above using the same rule and so on. This motivates the following results. We define a submonoid of ${\mathbb{N}}$

$$S'=\{x+y-z~\vert~ x,y,z\in S, x\geq y\geq z\}$$
and define $S^n$ recurrently as follows:
• $S^0=S$,
• $S^{n+1}=(S^n)'$.
Lemma 1   Let $S$ be a numerical semigroup. Then there exists $k\in {\mathbb{N}}$ such that $S^k= {\rm Arf}(S)$.
As we pointed out before, every numerical semigroup admits a unique minimal system of generators, and we prove that every Arf numerical semigroup has a unique minimal system of generators. A binary tree is a rooted tree such that every vertex has at most two sons (see [9]). Now, we describe a recursive procedure that arranges the set of all Arf numerical semigroups in a binary tree whose root is ${\mathbb{N}}$. The idea is to learn how to construct new Arf numerical semigroups by adding or removing an element from a given Arf numerical semigroup. We have that, if $S$ is Arf numerical semigroup then $S\cup \{g(s)\}$ is again an Arf numerical semigroup. Otherwise, if $S$ is an Arf numerical semigroup and $A$ is its minimal Arf system of generators then, $a\in A$ if and only if $S\setminus\{a\}$ is an Arf numerical semigroup. These facts allow us to construct recursively from the Arf numerical semigroup ${\mathbb{N}}$ the set of all Arf numerical semigroups (see the figure).

Figure 1: The binary tree of Arf numerical semigroups


Every vertex in the tree has at most two sons: if $T$ is a son of $S$, then either $T=S\setminus\{{\rm g}(S)+1\}$ or $T=S\setminus\{{\rm g}(S)+2\}$. Now, we present an algorithmic procedure for computing, from a finite subset $X$ of ${\mathbb{N}}$ with $\gcd(X)=1$, the elements of ${\rm Arf}(X)$. There exist a similitude between the algorithm described here and Euclid's algorithm for computing gcd's. It turns out that finding the elements of ${\rm Arf}(X)$ is much easier than computing $\langle X\rangle$. The main idea for this is the following result.
Theorem 2   Let $m, r_1,\ldots, r_p$ be nonnegative integers with greatest common divisor one. Then

$${\rm Arf}(m,m+r_1,\ldots, m+r_p)=(m+{\rm Arf}(m,r_1,\ldots,r_p))\cup \{0\}.$$

Let $X\subseteq {\mathbb{N}}\setminus\{0\}$ be such that $\gcd(X)=1$. Define recursively the following sequence of subsets of ${\mathbb{N}}$:
• $A_1=X$,
• $A_{n+1}=(\{x-{\rm min}_\leq A_n~\vert~ x\in A_n\}\setminus\{0\})\cup \{{\rm min}_\leq A_n\}$.
As a consequence of Euclid's algorithm for the computation of $\gcd(X)$, we obtain that there exists $q={\rm min}_\leq \{k\in {\mathbb{N}}~\vert~ 1\in A_k\}$.
Theorem 3   Under the standing notation, we have that
$$0,{\rm min}_\leq A_1, {\rm min}_\leq A_1+{\rm min}_\leq A_2, \ldots, {\rm min}_\leq A_1+\cdots+ {\rm min}_\leq A_{q-1}$$
are the elements in ${\rm Arf}(X)$ that are less than or equal to ${\rm g}({\rm Arf}(X))+1$.
Example 4   Let us compute ${\rm Arf}(7,24,33)$.
$A_1=\{7,24,33\}$, ${\rm min}_\leq A_1=7$,
$A_2=\{7,17,26\}$, ${\rm min}_\leq A_2=7$,
$A_3=\{7,10,19\}$, ${\rm min}_\leq A_3=7$,
$A_4=\{7,3,12\}$, ${\rm min}_\leq A_4=3$,
$A_5=\{4,3,9\}$, ${\rm min}_\leq A_5=3$,
$A_6=\{1,3,6\}$,
whence ${\rm Arf}(7,24,33)=\{0,7,14,21,24,27,\to \}$.




Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.

 

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