  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YA  [13Xnumerical semigroup[113X is a subset of the set [22XN[122X of nonnegative integers that
  is  closed  under  addition, contains [22X0[122X and whose complement in [22XN[122X is finite.
  The  smallest  positive  integer  belonging  to a numerical semigroup is its
  [13Xmultiplicity[113X.[133X
  
  [33X[0;0YLet  [22XS[122X  be  a numerical semigroup and [22XA[122X be a subset of [22XS[122X. We say that [22XA[122X is a
  [13Xsystem  of  generators[113X  of  [22XS[122X  if [22XS={ k_1 a_1+⋯+ k_n a_n | n,k_1,...,k_n∈ N,
  a_1,...,a_n∈  A}[122X.  The  set  [22XA[122X  is a [13Xminimal system of generators[113X of [22XS[122X if no
  proper subset of [22XA[122X is a system of generators of [22XS[122X.[133X
  
  [33X[0;0YEvery numerical semigroup has a unique minimal system of generators. This is
  a  data  that can be used in order to uniquely define a numerical semigroup.
  Observe  that  since  the  complement of a numerical semigroup in the set of
  nonnegative  integers  is  finite,  this  implies  that  the greatest common
  divisor  of  the  elements  of  a  numerical  semigroup  is  1, and the same
  condition  must  be fulfilled by its minimal system of generators (or by any
  of its systems of generators).[133X
  
  [33X[0;0YGiven  a  numerical  semigroup  [22XS[122X  and  a  nonzero  element [22Xs[122X in it, one can
  consider  for every integer [22Xi[122X ranging from [22X0[122X to [22Xs-1[122X, the smallest element in
  [22XS[122X  congruent  with  [22Xi[122X  modulo  [22Xs[122X,  say  [22Xw(i)[122X  (this element exists since the
  complement  of  [22XS[122X  in  [22XN[122X  is  finite).  Clearly  [22Xw(0)=0[122X.  The  set [22XAp(S,s)={
  w(0),w(1),...,  w(s-1)}[122X is called the [13XApéry set[113X of [22XS[122X with respect to [22Xs[122X. Note
  that  a  nonnegative integer [22Xx[122X congruent with [22Xi[122X modulo [22Xs[122X belongs to [22XS[122X if and
  only  if  [22Xw(i)≤ x[122X. Thus the pair [22X(s, Ap(S,s))[122X fully determines the numerical
  semigroup  [22XS[122X  (and can be used to easily solve the membership problem to [22XS[122X).
  This  set  is  in  fact  one of the most powerfull tools known for numerical
  semigroups,  and  it  is  used  almost  everywhere  in  the  computation  of
  components  and  invariants associated to a numerical semigroup. Usually the
  element  [22Xs[122X  is taken to be the multiplicity, since in this way the resulting
  Apéry set is the smallest possible.[133X
  
  [33X[0;0YA  [13Xgap[113X  of a numerical semigroup [22XS[122X is a nonnegative integer not belonging to
  [22XS[122X.  The  set of gaps of [22XS[122X is usually denoted by [22XH(S)[122X, and clearly determines
  uniquely  [22XS[122X.  Note  that if [22Xx[122X is a gap of [22XS[122X, then so are all the nonnegative
  integers dividing it. Thus in order to describe [22XS[122X we do not need to know all
  its  gaps, but only those that are maximal with respect to the partial order
  induced by division in [22XN[122X. These gaps are called [13Xfundamental gaps[113X.[133X
  
  [33X[0;0YThe  largest nonnegative integer not belonging to a numerical semigroup [22XS[122X is
  the  [13XFrobenius  number[113X  of  [22XS[122X. If [22XS[122X is the set of nonnegative integers, then
  clearly its Frobenius number is [22X-1[122X, otherwise its Frobenius number coincides
  with  the  maximum  of  the  gaps  (or fundamental gaps) of [22XS[122X. The Frobenius
  number  plus one is known as the [13Xconductor[113X of the semigroup. In this package
  we refer to the elements in the semigroup that are less than or equal to the
  conductor  as  [13Xsmall  elements[113X  of  the  semigroup.  Observe  that  from the
  definition,  if  [22XS[122X is a numerical semigroup with Frobenius number [22Xf[122X, then [22Xf+
  N∖{0}⊆ S[122X. An integer [22Xz[122X is a [13Xpseudo-Frobenius number[113X of [22XS[122X if [22Xz+S∖{0}⊆ S[122X. Thus
  the  Frobenius  number of [22XS[122X is one of its pseudo-Frobenius numbers. The [13Xtype[113X
  of   a   numerical   semigroup   is  the  cardinality  of  the  set  of  its
  pseudo-Frobenius numbers.[133X
  
  [33X[0;0YThe  number  of  numerical  semigroups  having  a  given Frobenius number is
  finite.  The  elements  in this set of numerical semigroups that are maximal
  with  respect to set inclusion are precisely those numerical semigroups that
  cannot  be  expressed  as  intersection  of  two  other numerical semigroups
  containing  them  properly, and thus they are known as [13Xirreducible[113X numerical
  semigroups.  Clearly,  every  numerical  semigroup  is  the  intersection of
  (finitely many) irreducible numerical semigroups.[133X
  
  [33X[0;0YA  numerical  semigroup  [22XS[122X with Frobenius number [22Xf[122X is [13Xsymmetric[113X if for every
  integer  [22Xx[122X,  either  [22Xx∈  S[122X  or  [22Xf-x∈  S[122X.  The  set  of irreducible numerical
  semigroups  with  odd  Frobenius  number coincides with the set of symmetric
  numerical  semigroups. The numerical semigroup [22XS[122X is [13Xpseudo-symmetric[113X if [22Xf[122X is
  even and for every integer [22Xx[122X not equal to [22Xf/2[122X either [22Xx∈ S[122X or [22Xf-x∈ S[122X. The set
  of  irreducible numerical semigroups with even Frobenius number is precisely
  the  set  of  pseudo-symmetric  numerical  semigroups.  These two classes of
  numerical semigroups have been widely studied in the literature due to their
  nice  applications  in  Algebraic Geometry. This is probably one of the main
  reasons  that made people turn their attention on numerical semigroups again
  in   the   last   decades.   Symmetric  numerical  semigroups  can  be  also
  characterized  as  those  with  type  one,  and  pseudo-symmetric  numerical
  semigroups  are  those  numerical semigroups with type two and such that its
  pseudo-Frobenius  numbers  are its Frobenius number and its Frobenius number
  divided by two.[133X
  
  [33X[0;0YAnother  class  of  numerical  semigroups  that  catched  the  attention  of
  researchers working on Algebraic Geometry and Commutative Ring Theory is the
  class   of  numerical  semigroups  with  maximal  embedding  dimension.  The
  [13Xembedding  dimension[113X  of  a  numerical  semigroup  is the cardinality of its
  minimal  system  of generators. It can be shown that the embedding dimension
  is  at  most  the  multiplicity  of  the  numerical  semigroup. Thus [13Xmaximal
  embedding  dimension[113X numerical semigroups are those numerical semigroups for
  which  their  embedding dimension and multiplicity coincide. These numerical
  semigroups  have  nice  maximal  properties, not only (of course) related to
  their  embedding  dimension, but also by means of their presentations. Among
  maximal  embedding  dimension  there are two classes of numerical semigroups
  that  have  been  studied  due  to  the  connections with the equivalence of
  algebroid  branches. A numerical semigroup [22XS[122X is Arf if for every [22Xx≥ y≥ z∈ S[122X,
  then  [22Xx+y-z∈  S[122X;  and  it  is [13Xsaturated[113X if the following condition holds: if
  [22Xs,s_1,...,s_r∈  S[122X are such that [22Xs_i≤ s[122X for all [22Xi∈ {1,...,r}[122X and [22Xz_1,...,z_r∈
  Z[122X are such that [22Xz_1s_1+⋯+z_rs_r≥ 0[122X, then [22Xs+z_1s_1+⋯ +z_rs_r∈ S[122X.[133X
  
  [33X[0;0YIf  we  look  carefully  inside  the  set of fundamental gaps of a numerical
  semigroup,  we see that there are some fulfilling the condition that if they
  are  added to the given numerical semigroup, then the resulting set is again
  a  numerical  semigroup.  These  elements  are  called  [13Xspecial  gaps[113X of the
  numerical semigroup. A numerical semigroup other than the set of nonnegative
  integers is irreducible if and only if it has only a special gap.[133X
  
  [33X[0;0YThe inverse operation to the one described in the above paragraph is that of
  removing  an  element of a numerical semigroup. If we want the resulting set
  to  be a numerical semigroup, then the only thing we can remove is a minimal
  generator.[133X
  
  [33X[0;0YLet  [22Xa,b,c,d[122X  be positive integers such that [22Xa/b < c/d[122X, and let [22XI=[a/b,c/d][122X.
  Then  the  set  [22XS(I)=  N∩ ⋃_n≥ 0 n I[122X is a numerical semigroup. This class of
  numerical  semigroups  coincides with that of sets of solutions to equations
  of  the  form  [22XA  x  mod  B  ≤ C x[122X with [22XA,B,C[122X positive integers. A numerical
  semigroup in this class is said to be [13Xproportionally modular[113X.[133X
  
  [33X[0;0YA  sequence  of positive rational numbers [22Xa_1/b_1 < ⋯ < a_n/b_n[122X with [22Xa_i,b_i[122X
  positive  integers is a [13XBézout sequence[113X if [22Xa_i+1b_i - a_i b_i+1=1[122X for all [22Xi∈
  {1,...,n-1}[122X.  If  [22Xa/b=a_1/b_1  <  ⋯  <  a_n/b_n  =c/d[122X,  then  [22XS([a/b,c/d])=⟨
  a_1,...,a_n⟩[122X.  Bézout sequences are not only interesting for this fact, they
  have  shown  to  be  a  major  tool  in  the study of proportionally modular
  numerical semigroups.[133X
  
  [33X[0;0YIf  [22XS[122X  is  a  numerical  semigroup and [22Xk[122X is a positive integer, then the set
  [22XS/k={ x∈ N | kx∈ S}[122X is a numerical semigroup, known as the [13Xquotient[113X [22XS[122X by [22Xk[122X.[133X
  
  [33X[0;0YLet  [22Xm[122X  be a positive integer. A [13Xsubadditive[113X function with period [22Xm[122X is a map
  [22Xf:  N->  N[122X  such  that  [22Xf(0)=0[122X, [22Xf(x+y)≤ f(x)+f(y)[122X and [22Xf(x+m)=f(x)[122X. If [22Xf[122X is a
  subadditive  function with period [22Xm[122X, then the set [22XM_f={ x∈ N | f(x)≤ x}[122X is a
  numerical  semigroup.  Moreover,  every numerical semigroup is of this form.
  Thus  a  numerical  semigroup  can be given by a subadditive function with a
  given  period. If [22XS[122X is a numerical semigroup and [22Xs∈ S, snot=0[122X, and [22XAp(S,s)={
  w(0),w(1),...,  w(s-1)}[122X, then [22Xf(x)=w(x mod s)[122X is a subadditive function with
  period [22Xs[122X such that [22XM_f=S[122X.[133X
  
  [33X[0;0YLet  [22XS[122X  be  a  numerical  semigroup  generated by [22X{n_1,...,n_k}[122X. Then we can
  define  the following morphism (called sometimes the factorization morphism)
  by  [22Xφ:  N^k  ->  S,  φ(a_1,...,a_k)=a_1n_1+⋯+a_kn_k[122X.  If  [22Xσ[122X  is  the  kernel
  congruence of [22Xφ[122X (that is, [22Xaσ b[122X if [22Xφ(a)=φ(b)[122X), then [22XS[122X is isomorphic to [22XN^k/σ[122X.
  A  [13Xpresentation[113X  for  [22XS[122X is a system of generators (as a congruence) of [22Xσ[122X. If
  [22X{n_1,...,n_p}[122X is a minimal system of generators, then a [13Xminimal presentation[113X
  is  a  presentation  such that none of its proper subsets is a presentation.
  Minimal  presentations  of  numerical semigroups coincide with presentations
  with  minimal  cardinality, though in general these two concepts are not the
  same for an arbitrary commutative semigroup.[133X
  
  [33X[0;0YA set [22XI[122X of integers is an [13Xideal relative to a numerical semigroup[113X [22XS[122X provided
  that  [22XI+S⊆ I[122X and that there exists [22Xd∈ S[122X such that [22Xd+I⊆ S[122X. If [22XI⊆ S[122X, we simply
  say that [22XI[122X is an [13Xideal[113X of [22XS[122X. If [22XI[122X and [22XJ[122X are relative ideals of [22XS[122X, then so is
  [22XI-J={z∈  Z  |  z+J⊆  I}[122X,  and  it is tightly related to the operation ":" of
  ideals in a commutative ring.[133X
  
  [33X[0;0YIn  this  package  we have implemented the functions needed to deal with the
  elements exposed in this introduction.[133X
  
  [33X[0;0YMany of the algorithms, and the necessary background to understand them, can
  be  found  in  the  monograph  [RGS09]. Some examples in this book have been
  illustrated with the help of this package. So the reader can also find there
  more examples on the usage of the functions implemented here.[133X
  
  [33X[0;0YThis package was presented in [DGSM06]. For a survey of the features of this
  package, see [DGS16].[133X
  
