GAP (NumericalSgps) - Chapter 11: Order induced by a numerical semigroup

Every numerical semigroup \(S\) induces an order on the set of integers: \(a \leq_S b\) if \(b - a \in S\). The fact that \(\le_S\) is a (partial) order follows from the fact that \(S\) is cancellative and reduced (the only unit is zero) monoid.

11.1 Posets induced by numerical semigroups

11.1-1 PosetNS

‣ PosetNS( S, A ) ( operation )

‣ PosetNS( A, S ) ( operation )

S is a numerical semigroup and A is a set of integers. Returns the poset with ground set A and order defined by the binary relation induced by S: \(a\preceq b\) if \(b - a\) in S.

gap> l:=[1..10];;
gap> s:=NumericalSemigroup(3,5,8);;
gap> p:=PosetNS(l,s);
<Poset defined wrt to numerical semigroup>

11.1-2 MaximalElements

‣ MaximalElements( P ) ( operation )

P is a poset induced by a numerical semigroup. Returns the list of maximal elements of P.

gap> s:=NumericalSemigroup(3,5,8);;
gap> l:=[1..10];;
gap> p:=PosetNS(l,s);;
gap> MaximalElements(p);
[ 10, 9, 8 ]
gap> Type(s)=Length(MaximalElements(PosetNS(AperyList(s),s)));
true

11.1-3 MinimalElements

‣ MinimalElements( P ) ( operation )

P is a poset induced by a numerical semigroup. Returns the list of minimal elements of P.

gap> s:=NumericalSemigroup(3,5,8);;
gap> l:=[1..10];;
gap> p:=PosetNS(l,s);;
gap> MinimalElements(p);
[ 1, 2, 3 ]

11.1-4 Upset

‣ Upset( P, l ) ( operation )

P is a poset induced by a numerical semigroup, l is a list of integers (contained in the ground set of P). Returns the upset of the list l in the poset P, that is, all elements of P greater than or equal to some element of l.

gap> s:=NumericalSemigroup(3,5,7);;
gap> l:=[1..10];;
gap> p:=PosetNS(l,s);;
gap> Upset(p,[2,4]);
[ 2, 4, 5, 7, 8, 9, 10 ]
gap> Upset(p,[2])=Filtered(l,i->i-2 in s);
true

11.1-5 Downset

‣ Downset( P, l ) ( operation )

P is a poset induced by a numerical semigroup, l is a list of integers (contained in the ground set of P). Returns the downset of the list l in the poset P, that is, all elements of P less than or equal to some element of l.

gap> s:=NumericalSemigroup(3,5,7);;
gap> l:=[1..10];;
gap> p:=PosetNS(l,s);;
gap> Downset(p,[5,6]);
[ 1, 2, 3, 5, 6 ]
gap> p:=PosetNS(s,AperyList(s));;
gap> Downset(p,Multiplicity(s)+PseudoFrobenius(s))=GroundSet(p);
true

11.1-6 AntichainsOfNumericalSemigroup

‣ AntichainsOfNumericalSemigroup( S, A ) ( function )

S is a numerical semigroup and A is a set of integers. Returns the set of antichains (sets of non-comparable elements) of A with respect to the ordering \(a\preceq b\) if \(b - a\) in S.

gap> s:=NumericalSemigroup(3,5,7);;
gap> AntichainsOfNumericalSemigroup(s,Gaps(s));
[ [  ], [ 4 ], [ 2 ], [ 2, 4 ], [ 1 ], [ 1, 2 ] ]

11.1-7 Antichains

‣ Antichains( P ) ( operation )

P is a poset defined by a numerical semigroup. Returns the set of antichains (sets of non-comparable elements) of P.

gap> s:=NumericalSemigroup(3,5,7);;
gap> p:=PosetNS(s,Gaps(s));;
gap> Antichains(p);
[ [  ], [ 4 ], [ 2 ], [ 2, 4 ], [ 1 ], [ 1, 2 ] ]
gap> Antichains(p)=AntichainsOfNumericalSemigroup(s,Gaps(s));
true

11.2 Hasse diagrams related to numerical semigroups

11.2-1 HasseDiagramOfNumericalSemigroup

‣ HasseDiagramOfNumericalSemigroup( S, A ) ( function )

S is a numerical semigroup and A is a set of integers. Returns a binary relation which is the Hasse diagram of A with respect to the ordering \(a\preceq b\) if \(b - a\) in S.

gap> s:=NumericalSemigroup(3,5,7);;
gap> HasseDiagramOfNumericalSemigroup(s,[1,2,3]);
<general mapping: <object> -> <object> >

11.2-2 HasseDiagram

‣ HasseDiagram( P ) ( operation )

P is a posed defined by a numerical semigroup. Returns a binary relation which is the Hasse diagram of P.

gap> s:=NumericalSemigroup(3,5,7);;
gap> p:=PosetNS(s,Gaps(s));;
gap> HasseDiagram(p)=HasseDiagramOfNumericalSemigroup(s,Gaps(s));
true

11.2-3 HasseDiagramOfBettiElementsOfNumericalSemigroup

‣ HasseDiagramOfBettiElementsOfNumericalSemigroup( S ) ( function )

S is a numerical semigroup. Applies HasseDiagramOfBettiElementsOfNumericalSemigroup with arguments S and its Betti elements.

gap> s:=NumericalSemigroup(3,5,7);;
gap> HasseDiagramOfBettiElementsOfNumericalSemigroup(s);
<general mapping: <object> -> <object> >

11.2-4 HasseDiagramOfAperyListOfNumericalSemigroup

‣ HasseDiagramOfAperyListOfNumericalSemigroup( S[, n] ) ( function )

S is a numerical semigroup, n is an integer (optional, if not provided, the multiplicity of the semigroup is taken as its value). Applies HasseDiagramOfBettiElementsOfNumericalSemigroup (11.2-3) with arguments S and the Apéry set of S with respect to n.

gap> s:=NumericalSemigroup(3,5,7);;
gap> HasseDiagramOfAperyListOfNumericalSemigroup(s);
<general mapping: <object> -> <object> >
gap> HasseDiagramOfAperyListOfNumericalSemigroup(s,10);
<general mapping: <object> -> <object> >

11 Order induced by a numerical semigroup