Posets

CHANGES

@@ -39,6 +39,12 @@
 - Added AntichainsOfNumericalSemigroup which computes antichains of sets of integers wrt order induced by a numerical semigroup
 - Added AtomMonoid for numerical sets
 - Added AssociatedNumericalSets for numerical semigroups
+- Added posets associated to numerical semigroups (PosetNS)
+- Added MinimalElements and MaximalElements for posets associated to numerical semigroups
+- Added Upset and Downset for posets associated to numerical semigroups
+- Anitichains for posets, which is a synonym of AntichainsOfNumericalSemigroup
+- HasseDiagram for posets associated to numerical semigroups as synonym of HasseDiagramOfNumericalSemigroup
+
 1.3.1 -> 1.4.0
 - Added NumericalSemigroupsWithFrobeniusNumberPC
 - Added NumericalSemigroupsWithGenusPC

doc/order.xml

@@ -1,5 +1,119 @@
 <?xml version="1.0" encoding="UTF-8"?>
 
+<Section>
+    <Heading>
+      Posets induced by numerical semigroups
+    </Heading>
+
+<ManSection>
+  <Oper Name="PosetNS" Arg="S, A" Label="for numerical semigroup and list of integers"/>
+  <Oper Name="PosetNS" Arg="A, S" Label="for list of integers and numerical semigroup"/>
+  <Description>
+    <C>S</C> is a numerical semigroup and <C>A</C> is a set of integers. Returns the poset with ground set <C>A</C> and order defined by the binary relation induced by <C>S</C>: <M>a\preceq b</M> if <M>b - a</M> in <C>S</C>.
+<Example><![CDATA[
+gap> l:=[1..10];;
+gap> s:=NumericalSemigroup(3,5,8);;
+gap> p:=PosetNS(l,s);
+<Poset defined wrt to numerical semigroup>
+]]></Example>
+  </Description>
+</ManSection>
+
+
+<ManSection>
+  <Oper Name="MaximalElements" Arg="P" Label="for posets defined by numerical semigroups"/>
+  <Description>
+    <C>P</C> is a poset induced by a numerical semigroup. Returns the list of maximal elements of <C>P</C>.
+<Example><![CDATA[
+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
+]]></Example>
+  </Description>
+</ManSection>
+
+<ManSection>
+  <Oper Name="MinimalElements" Arg="P" Label="for posets defined by numerical semigroups"/>
+  <Description>
+    <C>P</C> is a poset induced by a numerical semigroup. Returns the list of minimal elements of <C>P</C>.
+<Example><![CDATA[
+gap> s:=NumericalSemigroup(3,5,8);;
+gap> l:=[1..10];;
+gap> p:=PosetNS(l,s);;
+gap> MinimalElements(p);
+[ 1, 2, 3 ]
+]]></Example>
+  </Description>
+</ManSection>
+
+
+<ManSection>
+  <Oper Name="Upset" Arg="P,l" Label="for posets defined by numerical semigroups"/>
+  <Description>
+    <C>P</C> is a poset induced by a numerical semigroup, <C>l</C> is a list of integers (contained in the ground set of <C>P</C>). Returns the upset of the list <C>l</C> in the poset <C>P</C>, that is, all elements of <C>P</C> greater than or equal to some element of <C>l</C>.
+<Example><![CDATA[
+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
+]]></Example>
+  </Description>
+</ManSection>
+
+<ManSection>
+  <Oper Name="Downset" Arg="P,l" Label="for posets defined by numerical semigroups"/>
+  <Description>
+    <C>P</C> is a poset induced by a numerical semigroup, <C>l</C> is a list of integers (contained in the ground set of <C>P</C>). Returns the downset of the list <C>l</C> in the poset <C>P</C>, that is, all elements of <C>P</C> less than or equal to some element of <C>l</C>.
+<Example><![CDATA[
+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
+]]></Example>
+  </Description>
+</ManSection>
+
+<ManSection>
+  <Func Name="AntichainsOfNumericalSemigroup" Arg="S, A"/>
+  <Description>
+    <C>S</C> is a numerical semigroup and <C>A</C> is a set of integers. Returns the set of antichains (sets of non-comparable elements) of <C>A</C> with respect to the ordering <M>a\preceq b</M> if <M>b - a</M> in <C>S</C>.
+<Example><![CDATA[
+gap> s:=NumericalSemigroup(3,5,7);;
+gap> AntichainsOfNumericalSemigroup(s,Gaps(s));
+[ [  ], [ 4 ], [ 2 ], [ 2, 4 ], [ 1 ], [ 1, 2 ] ]
+]]></Example>
+  </Description>
+</ManSection>
+
+<ManSection>
+  <Oper Name="Antichains" Arg="P"/>
+  <Description>
+    <C>P</C> is a poset defined by a numerical semigroup. Returns the set of antichains (sets of non-comparable elements) of <C>P</C>.
+<Example><![CDATA[
+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
+]]></Example>
+  </Description>
+</ManSection>
+
+
+</Section>
+
 <Section>
     <Heading>
       Hasse diagrams related to numerical semigroups
@@ -17,6 +131,19 @@ gap> HasseDiagramOfNumericalSemigroup(s,[1,2,3]);
   </Description>
 </ManSection>
 
+<ManSection>
+  <Oper Name="HasseDiagram" Arg="P"/>
+  <Description>
+    <C>P</C> is a posed defined by a numerical semigroup. Returns a binary relation which is the Hasse diagram of <C>P</C>.
+<Example><![CDATA[
+gap> s:=NumericalSemigroup(3,5,7);;
+gap> p:=PosetNS(s,Gaps(s));;
+gap> HasseDiagram(p)=HasseDiagramOfNumericalSemigroup(s,Gaps(s));
+true
+]]></Example>
+  </Description>
+</ManSection>
+
 
 <ManSection>
   <Func Name="HasseDiagramOfBettiElementsOfNumericalSemigroup" Arg="S"/>
@@ -46,20 +173,3 @@ gap> HasseDiagramOfAperyListOfNumericalSemigroup(s,10);
 
 </Section>
 
-<Section>
-    <Heading>
-      Antichains
-    </Heading>
-<ManSection>
-  <Func Name="AntichainsOfNumericalSemigroup" Arg="S, A"/>
-  <Description>
-    <C>S</C> is a numerical semigroup and <C>A</C> is a set of integers. Returns the set of antichains (sets of non-comparable elements) of <C>A</C> with respect to the ordering <M>a\preceq b</M> if <M>b - a</M> in <C>S</C>.
-<Example><![CDATA[
-gap> s:=NumericalSemigroup(3,5,7);;
-gap> AntichainsOfNumericalSemigroup(s,Gaps(s));
-[ [  ], [ 4 ], [ 2 ], [ 2, 4 ], [ 1 ], [ 1, 2 ] ]
-]]></Example>
-  </Description>
-</ManSection>
-
-</Section>

gap/numset.gi

@@ -972,7 +972,7 @@ end);
 
 InstallMethod(FerrersDiagram, [IsNumericalSemigroup],
 function(s)
-  FerrersDiagram(AsNumericalSet(s));
+  return FerrersDiagram(AsNumericalSet(s));
 end);
 
 ###############################################################################
@@ -1000,13 +1000,13 @@ end);
 ###############################################################################
 InstallMethod(AssociatedNumericalSets, [IsNumericalSemigroup],
 function(S)
-     local F, gaps, H, H_Set, PF, PF_max, Tr, R, UpSet, DownSet, P, TrP,N, P_max, x, y, lim, 
-     A, C_A, Car_A, a, G, B1, B2, Z, L_anti, anti, Y, T, PF_DownSets, tuple, Explore;
+     local F, gaps, H, H_Set, PF, PF_max, Tr, R, Upset, Downset, P, TrP,N, P_max, x, y, lim, 
+     A, C_A, Car_A, a, G, B1, B2, Z, L_anti, anti, Y, T, PF_Downsets, tuple, Explore;
 
     # AUXILIARY FUNCTIONS
 
-    # UpSet(val): Returns a list of elements in Holes(S) >= val respecting S order
-     UpSet := function(val)
+    # Upset(val): Returns a list of elements in Holes(S) >= val respecting S order
+     Upset := function(val)
         local geq, z, start_index;
         geq := [];
         start_index := PositionSorted(H, val);
@@ -1016,8 +1016,8 @@ function(S)
         return geq;
     end;
 
-    # DownSet(val): Returns a list of elements in Holes(S) <= val respecting S order
-    DownSet := function(val)
+    # Downset(val): Returns a list of elements in Holes(S) <= val respecting S order
+    Downset := function(val)
         local leq, z;
         leq := [];
         for z in H do
@@ -1036,28 +1036,28 @@ function(S)
     PF := PseudoFrobenius(S);
     PF_max := Difference(PF, [F]);
 
-    # Pre-compute DownSets for all Pseudo-Frobenius numbers
-    PF_DownSets := rec();
+    # Pre-compute Downsets for all Pseudo-Frobenius numbers
+    PF_Downsets := rec();
     for P in PF do
-        PF_DownSets.(String(P)) := DownSet(P); 
+        PF_Downsets.(String(P)) := Downset(P); 
     od;
 
     # Pre-compute Frobenius Triangles for all Pseudo-Frobenius except F
     Tr := rec();
     N := Length(PF_max);
     if N > 0 then
         P_max:= PF_max[N]; 
-        Tr.(String(P_max)) := [ [P_max, F-P_max, 0, UpSet(F-P_max), [] ] ];  # The largest Pseudo-Frobenius after F(S) has no valid triangles except base case
+        Tr.(String(P_max)) := [ [P_max, F-P_max, 0, Upset(F-P_max), [] ] ];  # The largest Pseudo-Frobenius after F(S) has no valid triangles except base case
         for P in PF_max{[1..N-1]} do
-            TrP := [ [P, F-P, 0, UpSet(F-P), [] ] ]; # Base case
+            TrP := [ [P, F-P, 0, Upset(F-P), [] ] ]; # Base case
             lim := PositionSorted(H, F-P);
             if lim > Length(H) or H[lim] >= (F-P) then 
                 lim := lim - 1; 
             fi;
             for x in H{[1..lim]} do
                 y := F-P-x;
                 if y in H_Set then 
-                    Add(TrP, [P, x, y, UpSet(x), DownSet(F-y)]);
+                    Add(TrP, [P, x, y, Upset(x), Downset(F-y)]);
                 fi;
             od;
             Tr.(String(P)) := TrP;
@@ -1102,7 +1102,7 @@ function(S)
 
         # BRANCH 2: Exclude P from A
         new_B2 := ShallowCopy(B2_act);
-        UniteSet(new_B2, PF_DownSets.(String(P))); # Everything below P must be excluded
+        UniteSet(new_B2, PF_Downsets.(String(P))); # Everything below P must be excluded
         Explore(i+1, G_act, B1_act, new_B2, A_act);
     end;
 

gap/order.gd

@@ -1,3 +1,99 @@
+#############################################################################
+##
+#W  order.gd           Manuel Delgado <mdelgado@fc.up.pt>
+#W                          Pedro A. Garcia-Sanchez <pedro@ugr.es>
+##
+##
+#Y  Copyright 2026 by Manuel Delgado and Pedro Garcia-Sanchez 
+#Y  We adopt the copyright regulations of GAP as detailed in the
+#Y  copyright notice in the GAP manual.
+##
+#############################################################################
+
+
+#############################################################################
+##
+#R  IsPosetNSRep
+##
+##  The representation of a poset defined by a numerical semigroup and a list
+##  of integers.
+##
+#############################################################################
+DeclareRepresentation("IsPosetNSRep", IsAttributeStoringRep, []);
+
+#############################################################################
+##
+#C  IsPosetNS
+##
+##  The category of posets defined by numerical semigroups and lists of integers.
+##
+#############################################################################
+DeclareCategory( "IsPosetNS", IsPosetNSRep);
+
+# Elements of posets defined by numerical semigroups are integers, so posets 
+# defined by numerical semigroups are collections of integers.
+BindGlobal( "PosetNSType", NewType( CollectionsFamily(CyclotomicsFamily), IsPosetNS));
+
+#############################################################################
+##
+#F PosetNS(l,S)
+##
+## l is a list of integers and S a numerical semigroup
+##
+## returns the poset whose underlying set is l and the order is defined by
+## a <= b if b - a in S
+##
+#############################################################################
+DeclareOperation("PosetNS",[IsList, IsNumericalSemigroup]);
+# we allow changing the order of the arguments
+DeclareOperation("PosetNS",[IsNumericalSemigroup,IsList]);
+
+
+DeclareAttribute("UnderlyingNSPoset", IsPosetNS);
+DeclareAttribute("GroundSet", IsPosetNS);
+
+#############################################################################
+##
+#A MaximalElements(p)
+## Returns the list of maximal elements of the poset p
+##
+#############################################################################
+DeclareAttribute("MaximalElements", IsPosetNS);
+
+#############################################################################
+##
+#A MinimalElements(p)
+## Returns the list of minimal elements of the poset p
+##
+#############################################################################
+DeclareAttribute("MinimalElements", IsPosetNS);
+
+#############################################################################
+##
+#O Upset(p,l)
+##  Returns the upset of the list l in the poset p, that is, the set of 
+##  elements greater than or equal to some element of l.
+##
+#############################################################################
+DeclareOperation("Upset", [IsPosetNS, IsList]);
+
+#############################################################################
+##
+#O Downset(p,l)
+##  Returns the downset of the list l in the poset p, that is, the set of 
+##  elements less than or equal to some element of l.
+##
+#############################################################################
+DeclareOperation("Downset", [IsPosetNS, IsList]);
+
+#############################################################################
+##
+#O Antichains(p)
+##  Returns the set of antichains (sets of non comparable elements) of p.
+##
+#############################################################################
+DeclareOperation("Antichains", [IsPosetNS]);
+
 ############################################################################
 ##
 #F HasseDiagramOfNumericalSemigroup(s, A)
@@ -37,3 +133,12 @@ DeclareGlobalFunction("HasseDiagramOfAperyListOfNumericalSemigroup");
 ##
 ############################################################################
 DeclareGlobalFunction("AntichainsOfNumericalSemigroup");
+
+
+############################################################################
+##
+#O HasseDiagram(P)
+##  Returns the Hasse diagram of the poset P.
+##
+############################################################################
+DeclareOperation("HasseDiagram", [IsPosetNS]);