  
  [1X11 [33X[0;0YOrder induced by a numerical semigroup[133X[101X
  
  [33X[0;0YEvery numerical semigroup [22XS[122X induces an order on the set of integers: [22Xa ≤_S b[122X
  if  [22Xb  - a ∈ S[122X. The fact that [22X≤_S[122X is a (partial) order follows from the fact
  that [22XS[122X is cancellative and reduced (the only unit is zero) monoid.[133X
  
  
  [1X11.1 [33X[0;0YPosets induced by numerical semigroups[133X[101X
  
  [1X11.1-1 PosetNS[101X
  
  [33X[1;0Y[29X[2XPosetNS[102X( [3XS[103X, [3XA[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XPosetNS[102X( [3XA[103X, [3XS[103X ) [32X operation[133X
  
  [33X[0;0Y[10XS[110X  is  a  numerical  semigroup and [10XA[110X is a set of integers. Returns the poset
  with  ground  set  [10XA[110X  and order defined by the binary relation induced by [10XS[110X:
  [22Xapreceq b[122X if [22Xb - a[122X in [10XS[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xl:=[1..10];;[127X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,8);;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(l,s);[127X[104X
    [4X[28X<Poset defined wrt to numerical semigroup>[128X[104X
  [4X[32X[104X
  
  [1X11.1-2 MaximalElements[101X
  
  [33X[1;0Y[29X[2XMaximalElements[102X( [3XP[103X ) [32X operation[133X
  
  [33X[0;0Y[10XP[110X  is  a poset induced by a numerical semigroup. Returns the list of maximal
  elements of [10XP[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,8);;[127X[104X
    [4X[25Xgap>[125X [27Xl:=[1..10];;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(l,s);;[127X[104X
    [4X[25Xgap>[125X [27XMaximalElements(p);[127X[104X
    [4X[28X[ 10, 9, 8 ][128X[104X
    [4X[25Xgap>[125X [27XType(s)=Length(MaximalElements(PosetNS(AperyList(s),s)));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X11.1-3 MinimalElements[101X
  
  [33X[1;0Y[29X[2XMinimalElements[102X( [3XP[103X ) [32X operation[133X
  
  [33X[0;0Y[10XP[110X  is  a poset induced by a numerical semigroup. Returns the list of minimal
  elements of [10XP[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,8);;[127X[104X
    [4X[25Xgap>[125X [27Xl:=[1..10];;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(l,s);;[127X[104X
    [4X[25Xgap>[125X [27XMinimalElements(p);[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
  [4X[32X[104X
  
  [1X11.1-4 Upset[101X
  
  [33X[1;0Y[29X[2XUpset[102X( [3XP[103X, [3Xl[103X ) [32X operation[133X
  
  [33X[0;0Y[10XP[110X  is  a  poset  induced  by  a numerical semigroup, [10Xl[110X is a list of integers
  (contained  in  the ground set of [10XP[110X). Returns the upset of the list [10Xl[110X in the
  poset [10XP[110X, that is, all elements of [10XP[110X greater than or equal to some element of
  [10Xl[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xl:=[1..10];;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(l,s);;[127X[104X
    [4X[25Xgap>[125X [27XUpset(p,[2,4]);[127X[104X
    [4X[28X[ 2, 4, 5, 7, 8, 9, 10 ][128X[104X
    [4X[25Xgap>[125X [27XUpset(p,[2])=Filtered(l,i->i-2 in s);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X11.1-5 Downset[101X
  
  [33X[1;0Y[29X[2XDownset[102X( [3XP[103X, [3Xl[103X ) [32X operation[133X
  
  [33X[0;0Y[10XP[110X  is  a  poset  induced  by  a numerical semigroup, [10Xl[110X is a list of integers
  (contained in the ground set of [10XP[110X). Returns the downset of the list [10Xl[110X in the
  poset [10XP[110X, that is, all elements of [10XP[110X less than or equal to some element of [10Xl[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xl:=[1..10];;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(l,s);;[127X[104X
    [4X[25Xgap>[125X [27XDownset(p,[5,6]);[127X[104X
    [4X[28X[ 1, 2, 3, 5, 6 ][128X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(s,AperyList(s));;[127X[104X
    [4X[25Xgap>[125X [27XDownset(p,Multiplicity(s)+PseudoFrobenius(s))=GroundSet(p);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X11.1-6 AntichainsOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XAntichainsOfNumericalSemigroup[102X( [3XS[103X, [3XA[103X ) [32X function[133X
  
  [33X[0;0Y[10XS[110X  is  a  numerical semigroup and [10XA[110X is a set of integers. Returns the set of
  antichains  (sets  of  non-comparable  elements)  of  [10XA[110X  with respect to the
  ordering [22Xapreceq b[122X if [22Xb - a[122X in [10XS[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27XAntichainsOfNumericalSemigroup(s,Gaps(s));[127X[104X
    [4X[28X[ [  ], [ 4 ], [ 2 ], [ 2, 4 ], [ 1 ], [ 1, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X11.1-7 Antichains[101X
  
  [33X[1;0Y[29X[2XAntichains[102X( [3XP[103X ) [32X operation[133X
  
  [33X[0;0Y[10XP[110X is a poset defined by a numerical semigroup. Returns the set of antichains
  (sets of non-comparable elements) of [10XP[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(s,Gaps(s));;[127X[104X
    [4X[25Xgap>[125X [27XAntichains(p);[127X[104X
    [4X[28X[ [  ], [ 4 ], [ 2 ], [ 2, 4 ], [ 1 ], [ 1, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XAntichains(p)=AntichainsOfNumericalSemigroup(s,Gaps(s));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X11.2 [33X[0;0YHasse diagrams related to numerical semigroups[133X[101X
  
  [1X11.2-1 HasseDiagramOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XHasseDiagramOfNumericalSemigroup[102X( [3XS[103X, [3XA[103X ) [32X function[133X
  
  [33X[0;0Y[10XS[110X  is  a  numerical  semigroup  and [10XA[110X is a set of integers. Returns a binary
  relation  which  is  the  Hasse  diagram  of  [10XA[110X with respect to the ordering
  [22Xapreceq b[122X if [22Xb - a[122X in [10XS[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27XHasseDiagramOfNumericalSemigroup(s,[1,2,3]);[127X[104X
    [4X[28X<general mapping: <object> -> <object> >[128X[104X
  [4X[32X[104X
  
  [1X11.2-2 HasseDiagram[101X
  
  [33X[1;0Y[29X[2XHasseDiagram[102X( [3XP[103X ) [32X operation[133X
  
  [33X[0;0Y[10XP[110X  is  a  posed  defined by a numerical semigroup. Returns a binary relation
  which is the Hasse diagram of [10XP[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27Xp:=PosetNS(s,Gaps(s));;[127X[104X
    [4X[25Xgap>[125X [27XHasseDiagram(p)=HasseDiagramOfNumericalSemigroup(s,Gaps(s));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X11.2-3 HasseDiagramOfBettiElementsOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XHasseDiagramOfBettiElementsOfNumericalSemigroup[102X( [3XS[103X ) [32X function[133X
  
  [33X[0;0Y[10XS[110X          is         a         numerical         semigroup.         Applies
  [2XHasseDiagramOfBettiElementsOfNumericalSemigroup[102X  with  arguments  [10XS[110X  and its
  Betti elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27XHasseDiagramOfBettiElementsOfNumericalSemigroup(s);[127X[104X
    [4X[28X<general mapping: <object> -> <object> >[128X[104X
  [4X[32X[104X
  
  [1X11.2-4 HasseDiagramOfAperyListOfNumericalSemigroup[101X
  
  [33X[1;0Y[29X[2XHasseDiagramOfAperyListOfNumericalSemigroup[102X( [3XS[103X[, [3Xn[103X] ) [32X function[133X
  
  [33X[0;0Y[10XS[110X  is a numerical semigroup, [10Xn[110X is an integer (optional, if not provided, the
  multiplicity   of   the   semigroup   is   taken   as  its  value).  Applies
  [2XHasseDiagramOfBettiElementsOfNumericalSemigroup[102X  ([14X11.2-3[114X)  with  arguments [10XS[110X
  and the Apéry set of [10XS[110X with respect to [10Xn[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:=NumericalSemigroup(3,5,7);;[127X[104X
    [4X[25Xgap>[125X [27XHasseDiagramOfAperyListOfNumericalSemigroup(s);[127X[104X
    [4X[28X<general mapping: <object> -> <object> >[128X[104X
    [4X[25Xgap>[125X [27XHasseDiagramOfAperyListOfNumericalSemigroup(s,10);[127X[104X
    [4X[28X<general mapping: <object> -> <object> >[128X[104X
  [4X[32X[104X
  
