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A **numerical semigroup** is a subset of
that is closed under
addition, and generates
as a group (
and
denote the
set of integers and nonnegative integers, respectively). It is well known (see
for instance [2,8,13]) that if is a numerical semigroup,
then the set
has finitely many elements. The greatest integer
not belonging to is called the **Frobenius number** of , usually
denoted by
. Moreover, admits a unique minimal system of generators
(that is,
and no proper subset of
generates ). The
integers and are known as the **multiplicity** and
**embedding dimension** of , and they are denoted by
and , respectively. For a given
, we will denote by
the submonoid of
generated by . The monoid
is a numerical semigroup if and only if ( 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 is an **Arf numerical semigroup** if for every
such that
, we have that
(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
() as the intersection of all Arf numerical semigroups containing
(and thus
), and will denote it by
. If
, we say that is an **Arf system
of generators** of , and we will say that is **minimal** if no
proper subset of is an Arf system of generators of . For a numerical
semigroup ,
will be also called the **Arf closure** of .
Observe that if we are given
with , then
must contain the set of all the elements
of the form with
and
. 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

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*.