SingularClosurePartitionSemigroup

PackageInfo.g

@@ -303,9 +303,10 @@ Dependencies := rec(
   GAP := ">=4.9.0",
   NeededOtherPackages := [["orb", ">=4.8.2"],
                           ["io", ">=4.5.1"],
-                          ["digraphs", ">=1.0.0"],
+                          ["digraphs", ">=1.3.0"],
                           ["genss", ">=1.6.5"],
-                          ["images", ">=1.3.0"]],
+                          ["images", ">=1.3.0"],
+                          ["datastructures", ">=0.2.5"]],
   SuggestedOtherPackages := [["gapdoc", ">=1.5.1"]],
 
   ExternalConditions := []),

doc/semitrans.xml

@@ -376,3 +376,15 @@ gap> Size(WW);
     </Description>
   </ManSection>
 <#/GAPDoc>
+
+<#GAPDoc Label="SingularClosurePartitionSemigroup">
+   <ManSection>
+    <Oper Name="SingularClosurePartitionSemigroupSemigroup " Arg="G,n"/>
+    <Returns>A Partition monoid which contains all singular Partitions.</Returns>
+    <Description>
+      When <A>G</A> is a subgroup of the symmertic group on <A>n</A> points then 
+	  this returns the least, by containment, Partition monoid of degree 
+          <A>n</A> which contains <A>G</A> and all singular Partitions.
+    </Description>
+  </ManSection>
+<#/GAPDoc>

gap/semigroups/semitrans.gd

@@ -43,3 +43,6 @@ DeclareAttribute("DigraphCore", IsDigraph);
 DeclareOperation("WreathProduct",
                  [IsMultiplicativeElementCollection,
                   IsMultiplicativeElementCollection]);
+
+DeclareOperation("SingularClosurePartitionSemigroup",
+                 [IsGroup, IsPosInt]);

gap/semigroups/semitrans.gi

@@ -691,6 +691,133 @@ function(M, S)
   return Semigroup(maps);
 end);
 
+InstallMethod(SingularClosurePartitionSemigroup,
+ " for permutation group and degree", [IsGroup, IsPosInt],
+function(G, n)
+  local D, id, f, o, O, d, I, J, K, L, N, e, q, i,
+  Digo, nlc, v, E, j, k, M, gcount, g;
+
+  # Digraph with nodes {1, .., n} and an edge from x to y if there exists g in
+  # G such that g(x) = y.
+  D := Digraph([1 .. n], {x, y} -> ForAny(G, g -> x ^ g = y));
+  # List such that if v is vertex in digraph D, then id[v] is the index of the
+  # strongly connected component of D containing v
+  id := DigraphStronglyConnectedComponents(D).id;
+  # Takes a pair [i, j] and returns the pair consisting of the index of the scc
+  # of D containing i and the index of the scc of D containing j.
+  f := pair -> [id[pair[1]], id[pair[2]]];
+  # Compute the orbits of G on pairs
+  o := Orbits(G, Combinations([1 .. n], 2), OnSets);
+  # List(o, x -> f(x[1])) apply the function f (above) to one representative
+  # from every orbit of o, and create the digraph with these edges.
+  O := DigraphByEdges(List(o, x -> f(x[1])));
+  # removes loops to count number of non loops
+  d := DigraphNrEdges(DigraphRemoveLoops(O));
+  # calculate number of digraph vertices
+  v := DigraphNrVertices(O);
+  g := Size(Generators(G));
+  # special case for if quoteient is a point
+  e := DigraphNrEdges(O);
+  N := [1 .. (2 * e) + g];
+  if v = 1 then
+    for i in [1 .. DigraphNrEdges(O)] do
+      K := o[i][1][1];
+      L := o[i][1][2];
+      N[2 * i] := AsBipartition(Transformation([K], [L]), n);
+    od;
+    # cases are if only 1 non loop and otherweise
+  elif d = 1 then
+    e := DigraphNrEdges(O);
+    N := [1 .. (2 * e) + 2 + g];
+    for i in [1 .. DigraphNrEdges(O)] do
+      if DigraphEdges(O)[i][1] = DigraphEdges(O)[i][2] then
+        K := o[i][1][1];
+        L := o[i][1][2];
+        N[2 * i] := AsBipartition(Transformation([K], [L]), n);
+      else
+         q := i;
+      fi;
+    od;
+    I := o[q][1][1];
+    J := o[q][1][2];
+    K := o[q][2][2];
+    N[2 * (q)] := AsBipartition(Transformation([J], [I]), n);
+    N[2 * e + 2] := AsBipartition(Transformation([I, J], [J, K]), n);
+    # second case as StrongOrientation function
+    # only works on graphs with more vertex than 2
+  elif d = 2 then
+    e := DigraphNrEdges(O);
+    N := [1 .. (2 * e) + g];
+    nlc := 0;
+    for i in [1 .. DigraphNrEdges(O)] do
+      if DigraphEdges(O)[i][1] = DigraphEdges(O)[i][2] then
+        K := o[i][1][1];
+        L := o[i][1][2];
+      elif nlc < 1 then
+        K := (o)[i][1][1];
+        L := (o)[i][1][2];
+        nlc := nlc + 1;
+      else
+        K := (o)[i][1][2];
+        L := (o)[i][1][1];
+      fi;
+      N[2 * i] := AsBipartition(Transformation([K], [L]), n);
+    od;
+  else
+    e := DigraphNrEdges(O);
+    N := [1 .. (2 * e) + g];
+    Digo := StrongOrientation(DigraphSymmetricClosure(O));
+    for i in [1 .. DigraphNrEdges(O)] do
+      if DigraphEdges(O)[i][1] = DigraphEdges(O)[i][2] then
+        K := o[i][1][1];
+        L := o[i][1][2];
+      elif DigraphEdges(O)[i][1] in DigraphEdges(Digo) then
+        K := o[i][1][1];
+        L := o[i][1][2];
+      else
+        K := o[i][1][2];
+        L := o[i][1][1];
+      fi;
+         # using these start and end point values construct
+         # a transfomarion which is constant on that edge
+      N[2 * i] := AsBipartition(Transformation([K], [L]), n);
+    od;
+  fi;
+  if d = 1 then
+    for i in [1 .. DigraphNrEdges(O) + 1] do
+      E := ExtRepOfObj(N[2 * i]);
+      M := [1 .. Size(E)];
+      for j in [1 .. Size(E)] do
+        M[j] := ShallowCopy(E[j]);
+      od;
+      for j in [1 .. Size(E)] do
+        for k in [1 .. Size(E[j])] do
+          M[j][k] := - E[j][k];
+        od;
+      od;
+      N[(2 * i) - 1] := Bipartition(M);
+    od;
+  else
+    for i in [1 .. DigraphNrEdges(O)] do
+      E := ExtRepOfObj(N[2 * i]);
+      M := [1 .. Size(E)];
+      for j in [1 .. Size(E)] do
+        M[j] := ShallowCopy(E[j]);
+      od;
+      for j in [1 .. Size(E)] do
+        for k in [1 .. Size(E[j])] do
+          M[j][k] := - E[j][k];
+        od;
+      od;
+      N[(2 * i) - 1] := Bipartition(M);
+    od;
+  fi;
+  for gcount in [1 .. Size(Generators(G))] do
+    N[Size(N) - gcount + 1] := AsBipartition(Generators(G)[gcount], n);
+  od;
+  return Semigroup(N);
+  end);
+
 #############################################################################
 # ?. Isomorphisms
 #############################################################################

scripts/travis-build-dependencies.sh

@@ -58,13 +58,10 @@ CURL="curl --connect-timeout 5 --max-time 10 --retry 5 --retry-delay 0 --retry-m
 
 ################################################################################
 # Install digraphs, genss, io, orb, images, and profiling
-PKGS=( "digraphs" "genss" "io" "orb" "images")
+PKGS=( "digraphs" "genss" "io" "orb" "images" "datastructures" )
 if [ "$SUITE"  == "coverage" ]; then
   PKGS+=( "profiling" )
 fi
-if [ "$PACKAGES" == "latest" ]; then 
-  PKGS+=( "datastructures" )
-fi
 
 for PKG in "${PKGS[@]}"; do
   cd $GAPROOT/pkg

tst/standard/semitrans.tst

@@ -2776,6 +2776,24 @@ true
 gap> Transformation([2, 1, 4, 3, 6, 5, 6, 5, 7, 8]) in WW;
 false
 
+# SingularClosurePartitionSemigroup
+gap> S := SingularClosurePartitionSemigroup(Group((1, 2)(3, 4)), 4);
+<bipartition semigroup of degree 4 with 9 generators>
+gap> Size(S) = Bell(2 * 4) - Factorial(4) + 2;
+true
+gap> S := SingularClosurePartitionSemigroup(Group((1, 2), (3, 4)), 4);
+<bipartition semigroup of degree 4 with 10 generators>
+gap> Size(S) = Bell(2 * 4) - Factorial(4) + 4;
+true
+gap> S := SingularClosurePartitionSemigroup(SymmetricGroup(4), 4);
+<bipartition semigroup of degree 4 with 4 generators>
+gap> Size(S) = Bell(2 * 4);
+true
+gap> S := SingularClosurePartitionSemigroup(Group((1, 2)), 4);
+<bipartition semigroup of degree 4 with 9 generators>
+gap> Size(S) = Bell(2 * 4) - Factorial(4) + 2;
+true
+
 # DigraphCore
 gap> D := CompleteBipartiteDigraph(4, 4);
 <immutable complete bipartite digraph with bicomponent sizes 4 and 4>