  
  [1X2 [33X[0;0YNumerical Semigroups[133X[101X
  
  [33X[0;0YThis chapter describes how to create numerical semigroups in [5XGAP[105X and perform
  some basic tests.[133X
  
  
  [1X2.1 [33X[0;0YGenerating Numerical Semigroups[133X[101X
  
  [33X[0;0YWe recall some definitions from Chapter [14X1[114X.[133X
  
  [33X[0;0YA  numerical semigroup 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.[133X
  
  [33X[0;0YWe  refer  to  the  elements  in a numerical semigroup that are less than or
  equal to the conductor as [13Xsmall elements[113X of the semigroup.[133X
  
  [33X[0;0YA  [13Xgap[113X  of a numerical semigroup [22XS[122X is a nonnegative integer not belonging to
  [22XS[122X. The [13Xfundamental gaps[113X of [22XS[122X are those gaps that are maximal with respect to
  the partial order induced by division in [22XN[122X.[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.[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. If [22XC = 1[122X, then
  it is said to be [13Xmodular[113X.[133X
  
  [33X[0;0YThere  are different ways to specify a numerical semigroup [22XS[122X, namely, by its
  generators;  by  its gaps, its fundamental or special gaps by its Apéry set,
  just to name some. In this section we describe functions that may be used to
  specify, in one of these ways, a numerical semigroup in [5XGAP[105X.[133X
  
  [1X2.1-1 NumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [[3XString[103X, ][3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroupByGenerators[102X( [3XList[103X ) [32X function[133X
  
  [33X[0;0Y[10XList[110X is a list of nonnegative integers with greatest common divisor equal to
  one.  These  integers  may be given as a list or by a sequence of individual
  elements. The output is the numerical semigroup spanned by [10XList[110X.[133X
  
  [33X[0;0Y[10XString[110X  does  not  need  to  be  present.  When  it  is  present, it must be
  [10X"generators"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs1 := NumericalSemigroup(3,5,7);               [127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs2 := NumericalSemigroup([3,5,7]);[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs3 := NumericalSemigroupByGenerators(3,5,7);             [127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs4 := NumericalSemigroupByGenerators([3,5,7]);[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs5 := NumericalSemigroup("generators",3,5,7); [127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs6 := NumericalSemigroup("generators",[3,5,7]);[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xs1=s2;s2=s3;s3=s4;s4=s5;s5=s6;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 NumericalSemigroupBySubAdditiveFunction[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupBySubAdditiveFunction[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0YA  periodic  subadditive function with period [22Xm[122X is given through the list of
  images of the integers from [22X1[122X to [22Xm[122X, [Ros07]. The image of [22Xm[122X has to be [22X0[122X. The
  output is the numerical semigroup determined by this subadditive function.[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"subadditive"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := NumericalSemigroupBySubAdditiveFunction([5,4,2,0]);[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27Xt := NumericalSemigroup("subadditive",[5,4,2,0]);;     [127X[104X
    [4X[25Xgap>[125X [27Xs=t;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.1-3 NumericalSemigroupByAperyList[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupByAperyList[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0Y[10XList[110X is an Apéry list. The output is the numerical semigroup whose Apéry set
  with respect to the length of given list is [10XList[110X.[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"apery"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,11);;[127X[104X
    [4X[25Xgap>[125X [27Xap := AperyListOfNumericalSemigroupWRTElement(s,20);[127X[104X
    [4X[28X[ 0, 21, 22, 3, 24, 25, 6, 27, 28, 9, 30, 11, 12, 33, 14, 15, 36, 17, 18, 39 ][128X[104X
    [4X[25Xgap>[125X [27Xt:=NumericalSemigroupByAperyList(ap);;[127X[104X
    [4X[25Xgap>[125X [27Xr := NumericalSemigroup("apery",ap);;[127X[104X
    [4X[25Xgap>[125X [27Xs=t;t=r;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.1-4 NumericalSemigroupBySmallElements[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupBySmallElements[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0Y[10XList[110X is the set of small elements of a numerical semigroup, that is, the set
  of  all elements not greater than the conductor. The output is the numerical
  semigroup with this set of small elements. When no such semigroup exists, an
  error is returned.[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"elements"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,11);;[127X[104X
    [4X[25Xgap>[125X [27Xse := SmallElements(s);[127X[104X
    [4X[28X[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ][128X[104X
    [4X[25Xgap>[125X [27Xt := NumericalSemigroupBySmallElements(se);;[127X[104X
    [4X[25Xgap>[125X [27Xr := NumericalSemigroup("elements",se);;[127X[104X
    [4X[25Xgap>[125X [27Xs=t;t=r;                                            [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xe := [ 0, 3, 6, 9, 11, 14, 15, 17, 18, 20 ];    [127X[104X
    [4X[28X[ 0, 3, 6, 9, 11, 14, 15, 17, 18, 20 ][128X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroupBySmallElements(e);[127X[104X
    [4X[28XError, The argument does not represent a numerical semigroup called from[128X[104X
    [4X[28X<function "NumericalSemigroupBySmallElements">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 35 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27X[127X[104X
  [4X[32X[104X
  
  [1X2.1-5 NumericalSemigroupByGaps[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupByGaps[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0Y[10XList[110X  is  the  set  of  gaps  of  a  numerical  semigroup. The output is the
  numerical semigroup with this set of gaps. When no semigroup exists with the
  given set as set of gaps, an error is returned.[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"gaps"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg := [ 1, 2, 4, 5, 7, 8, 10, 13, 16 ];;[127X[104X
    [4X[25Xgap>[125X [27Xs := NumericalSemigroupByGaps(g);;[127X[104X
    [4X[25Xgap>[125X [27Xt := NumericalSemigroup("gaps",g);;[127X[104X
    [4X[25Xgap>[125X [27Xs=t;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xh := [ 1, 2, 5, 7, 8, 10, 13, 16 ];;   [127X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroupByGaps(h);[127X[104X
    [4X[28XError, The argument does not represent the gaps of a numerical semigroup called[128X[104X
    [4X[28X from[128X[104X
    [4X[28X<function "NumericalSemigroupByGaps">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 34 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27X[127X[104X
  [4X[32X[104X
  
  [1X2.1-6 NumericalSemigroupByFundamentalGaps[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupByFundamentalGaps[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0Y[10XList[110X  is  the set of fundamental gaps of a numerical semigroup, [RGSGGJM04].
  The  output  is  the  numerical semigroup determined by these gaps. When the
  given  set  contains  elements (which will be gaps) that are not fundamental
  gaps, they are silently removed.[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"fundamentalgaps"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfg := [ 11, 14, 17, 20, 23, 26, 29, 32, 35 ];;[127X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroupByFundamentalGaps(fg);[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("fundamentalgaps",fg);     [127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27Xlast=last2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgg := [ 11, 17, 20, 22, 23, 26, 29, 32, 35 ];; #22 is not fundamental[127X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("fundamentalgaps",fg);     [127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
  [4X[32X[104X
  
  [1X2.1-7 NumericalSemigroupByAffineMap[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupByAffineMap[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X function[133X
  
  [33X[0;0YGiven three nonnegative integers [3Xa[103X, [3Xb[103X and [3Xc[103X, with [22Xa,c>0[122X and [22Xgcd(b,c)=1[122X, this
  function  returns  the least (with respect to set order inclusion) numerical
  semigroup  containing  [3Xc[103X  and closed under the map [22Xx↦ ax+b[122X. The procedure is
  explained in [Ugo16].[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"affinemap"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroupByAffineMap(3,1,3);[127X[104X
    [4X[28X<Numerical semigroup with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XSmallElements(s);[127X[104X
    [4X[28X[ 0, 3, 6, 9, 10, 12, 13, 15, 16, 18 ][128X[104X
    [4X[25Xgap>[125X [27Xt:=NumericalSemigroup("affinemap",3,1,3);;[127X[104X
    [4X[25Xgap>[125X [27Xs=t;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.1-8 ModularNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XModularNumericalSemigroup[102X( [3Xa[103X, [3Xb[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3Xa[103X, [3Xb[103X ) [32X function[133X
  
  [33X[0;0YGiven  two  positive  integers  [3Xa[103X  and  [3Xb[103X,  this  function returns a modular
  numerical semigroup satisfying [22Xax mod b le x[122X, [RGSUB05].[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"modular"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XModularNumericalSemigroup(3,7);[127X[104X
    [4X[28X<Modular numerical semigroup satisfying 3x mod 7 <= x >[128X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("modular",3,7);  [127X[104X
    [4X[28X<Modular numerical semigroup satisfying 3x mod 7 <= x >[128X[104X
  [4X[32X[104X
  
  [1X2.1-9 ProportionallyModularNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XProportionallyModularNumericalSemigroup[102X( [3Xa[103X, [3Xb[103X, [3Xc[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3Xa[103X, [3Xb[103X ) [32X function[133X
  
  [33X[0;0YGiven  three  positive  integers  [3Xa[103X,  [3Xb[103X  and  [3Xc[103X,  this  function  returns  a
  proportionally  modular  numerical  semigroup  satisfying  [22Xaxmod  b  le  cx[122X,
  [RAGGUB03].[133X
  
  [33X[0;0YIn the second form, [10XString[110X must be [10X"propmodular"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XProportionallyModularNumericalSemigroup(3,7,12);[127X[104X
    [4X[28X<Proportionally modular numerical semigroup satisfying 3x mod 7 <= 12x >[128X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("propmodular",3,7,12);[127X[104X
    [4X[28X<Proportionally modular numerical semigroup satisfying 3x mod 7 <= 12x >[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWhen [22Xc=1[122X, the semigroup is seen as a modular numerical semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("propmodular",67,98,1);[127X[104X
    [4X[28X<Modular numerical semigroup satisfying 67x mod 98 <= x >[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNumerical semigroups generated by an interval of positive integers are known
  to  be  proportionally  modular,  and  thus  they are treated as such, since
  membership   and   other  problems  can  be  solved  efficiently  for  these
  semigroups.[133X
  
  [1X2.1-10 NumericalSemigroupByInterval[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupByInterval[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0YThe  input  is  a  list  of rational numbers defining a closed interval. The
  output  is  the  semigroup  of  numerators  of  all rational numbers in this
  interval, [RAGGUB03].[133X
  
  [33X[0;0Y[10XString[110X  does  not  need  to  be  present.  When  it  is  present, it must be
  [10X"interval"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroupByInterval(7/5,5/3);[127X[104X
    [4X[28X<Proportionally modular numerical semigroup satisfying 25x mod 35 <= 4x >[128X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("interval",[7/5,5/3]);[127X[104X
    [4X[28X<Proportionally modular numerical semigroup satisfying 25x mod 35 <= 4x >[128X[104X
    [4X[25Xgap>[125X [27XSmallElements(last);[127X[104X
    [4X[28X[ 0, 3, 5 ][128X[104X
  [4X[32X[104X
  
  [1X2.1-11 NumericalSemigroupByOpenInterval[101X
  
  [33X[1;0Y[29X[2XNumericalSemigroupByOpenInterval[102X( [3XList[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumericalSemigroup[102X( [3XString[103X, [3XList[103X ) [32X function[133X
  
  [33X[0;0YThe  input  is  a  list  of  rational numbers defining an open interval. The
  output  is  the  semigroup  of  numerators  of  all rational numbers in this
  interval, [RUB06].[133X
  
  [33X[0;0Y[10XString[110X  does  not  need  to  be  present.  When  it  is  present, it must be
  [10X"openinterval"[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroupByOpenInterval(7/5,5/3);[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XNumericalSemigroup("openinterval",[7/5,5/3]);[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XSmallElements(last);                         [127X[104X
    [4X[28X[ 0, 3, 6, 8 ] [128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YSome basic tests[133X[101X
  
  [33X[0;0YThis  section  describes some basic tests on numerical semigroups. The first
  described  tests  refer  to what the semigroup is currently known to be (not
  necessarily the way it was created). Then are presented functions to test if
  a  given  list represents the small elements, gaps or the Apéry set (see [14X1.[114X)
  of  a  numerical  semigroup;  to  test  if an integer belongs to a numerical
  semigroup and if a numerical semigroup is a subsemigroup of another one.[133X
  
  [1X2.2-1 IsNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XIsNumericalSemigroup[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupByGenerators[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupByInterval[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupByOpenInterval[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupBySubAdditiveFunction[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupByAperyList[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupBySmallElements[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupByGaps[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsNumericalSemigroupByFundamentalGaps[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsProportionallyModularNumericalSemigroup[102X( [3XNS[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsModularNumericalSemigroup[102X( [3XNS[103X ) [32X attribute[133X
  
  [33X[0;0Y[3XNS[103X  is a numerical semigroup and these attributes are available (their names
  should be self explanatory).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,7);[127X[104X
    [4X[28X<Numerical semigroup with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XAperyListOfNumericalSemigroupWRTElement(s,30);;[127X[104X
    [4X[25Xgap>[125X [27Xt:=NumericalSemigroupByAperyList(last);[127X[104X
    [4X[28X<Numerical semigroup>[128X[104X
    [4X[25Xgap>[125X [27XIsNumericalSemigroupByGenerators(s);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsNumericalSemigroupByGenerators(t);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNumericalSemigroupByAperyList(s);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNumericalSemigroupByAperyList(t);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-2 RepresentsSmallElementsOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XRepresentsSmallElementsOfNumericalSemigroup[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YTests  if  the  list  [3XL[103X  (which  has to be a set) may represent the ``small"
  elements of a numerical semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ];[127X[104X
    [4X[28X[ 0, 3, 6, 9, 11, 12, 14, 15, 17, 18, 20 ][128X[104X
    [4X[25Xgap>[125X [27XRepresentsSmallElementsOfNumericalSemigroup(L);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XL:=[ 6, 9, 11, 12, 14, 15, 17, 18, 20 ];[127X[104X
    [4X[28X[ 6, 9, 11, 12, 14, 15, 17, 18, 20 ][128X[104X
    [4X[25Xgap>[125X [27XRepresentsSmallElementsOfNumericalSemigroup(L);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.2-3 RepresentsGapsOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XRepresentsGapsOfNumericalSemigroup[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YTests  if  the  list  [3XL[103X  may  represent  the  gaps  (see  [14X1.[114X) of a numerical
  semigroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,7);[127X[104X
    [4X[28X<Numerical semigroup with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XL:=GapsOfNumericalSemigroup(s);[127X[104X
    [4X[28X[ 1, 2, 4, 5, 8, 11 ][128X[104X
    [4X[25Xgap>[125X [27XRepresentsGapsOfNumericalSemigroup(L);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XL:=Set(List([1..21],i->RandomList([1..50])));[127X[104X
    [4X[28X[ 2, 6, 7, 8, 10, 12, 14, 19, 24, 28, 31, 35, 42, 50 ][128X[104X
    [4X[25Xgap>[125X [27XRepresentsGapsOfNumericalSemigroup(L);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.2-4 IsAperyListOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XIsAperyListOfNumericalSemigroup[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YTests  whether  a  list  [3XL[103X  of  integers  may  represent the Apéry list of a
  numerical  semigroup. It returns [9Xtrue[109X when the periodic function represented
  by  [3XL[103X is subadditive (see [2XRepresentsPeriodicSubAdditiveFunction[102X ([14XA.2-1[114X)) and
  the  remainder  of  the division of [10XL[i][110X by the length of [3XL[103X is [10Xi[110X and returns
  [9Xfalse[109X otherwise (the criterium used is the one explained in [Ros96b]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsAperyListOfNumericalSemigroup([0,21,7,28,14]);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-5 IsSubsemigroupOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XIsSubsemigroupOfNumericalSemigroup[102X( [3XS[103X, [3XT[103X ) [32X function[133X
  
  [33X[0;0Y[3XS[103X and [3XT[103X are numerical semigroups. Tests whether [3XT[103X is contained in [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := NumericalSemigroup("modular", 5,53);[127X[104X
    [4X[28X<Modular numerical semigroup satisfying 5x mod 53 <= x >[128X[104X
    [4X[25Xgap>[125X [27XT:=NumericalSemigroup(2,3);[127X[104X
    [4X[28X<Numerical semigroup with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroupOfNumericalSemigroup(T,S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSubsemigroupOfNumericalSemigroup(S,T);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X2.2-6 IsSubset[101X
  
  [33X[1;0Y[29X[2XIsSubset[102X( [3XS[103X, [3XT[103X ) [32X attribute[133X
  
  [33X[0;0Y[3XS[103X  is  a  numerical semigroup. [3XT[103X can be a numerical semigroup, in which case
  the   function  is  just  a  synonym  of  [2XIsSubsemigroupOfNumericalSemigroup[102X
  ([14X2.2-5[114X),  or a list of integers, in which case tests whether all elements of
  the list belong to [3XS[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xns1 := NumericalSemigroup(5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xns2 := NumericalSemigroup(5,7,11);;[127X[104X
    [4X[25Xgap>[125X [27XIsSubset(ns1,ns2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsSubset(ns2,[5,15]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSubset(ns1,[5,11]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsSubset(ns2,ns1);   [127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X2.2-7 BelongsToNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XBelongsToNumericalSemigroup[102X( [3Xn[103X, [3XS[103X ) [32X operation[133X
  [33X[1;0Y[29X[2X\in[102X( [3Xn[103X, [3XS[103X ) [32X operation[133X
  
  [33X[0;0Y[3Xn[103X  is  an integer and [3XS[103X is a numerical semigroup. Tests whether [3Xn[103X belongs to
  [3XS[103X.  [10X\in(n,S)[110X calls the infix variant [10Xn in S[110X, and both can be seen as a short
  for  [10XBelongsToNumericalSemigroup(n,S)[110X.  Several  methods are implemented for
  membership,  depending on the properties of [3XS[103X known. For instance, there are
  methods  if  any  of  the  following  information is known: Apéry set, small
  elements,    defining   (proportionally)   modular   Diophantine   equation,
  fundamental gaps, gaps, generators.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS := NumericalSemigroup("modular", 5,53);[127X[104X
    [4X[28X<Modular numerical semigroup satisfying 5x mod 53 <= x >[128X[104X
    [4X[25Xgap>[125X [27XBelongsToNumericalSemigroup(15,S);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27X15 in S;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XSmallElementsOfNumericalSemigroup(S);[127X[104X
    [4X[28X[ 0, 11, 12, 13, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 38, 39, 43 ][128X[104X
    [4X[25Xgap>[125X [27XBelongsToNumericalSemigroup(13,S);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X13 in S;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
