Add PowerSemigroup

doc/semicons.xml

@@ -355,6 +355,36 @@ true]]></Example>
   </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="PowerSemigroup">
+  <ManSection>
+    <Oper Name="PowerSemigroup" Arg="S"/>
+    <Returns>
+      A semigroup or monoid.
+    </Returns>
+    <Description>
+      The power semigroup <M>\mathcal{P}(S)</M> over the finite semigroup
+      <A>S</A> consists of the set of all non-empty subsets of <A>S</A>, along
+      with the binary operation of setwise multiplication, defined by
+      <Display>(X, Y) \mapsto \{xy : x \in X, y \in Y\}</Display>
+      <P/>
+      If <A>S</A> is a monoid, then <C>PowerSemigroup</C> returns a monoid.
+      <P/>
+      Power semigroups grow exponentially; for a semigroup <A>S</A> of size
+      <M>n</M>, the corresponding power semigroup has size <M>2^{n}-1</M>, so
+      its multiplication table may exhaust memory.
+
+      <Example><![CDATA[
+gap> S := SemigroupByMultiplicationTable([[1, 2], [2, 1]]);;
+gap> S1 := PowerSemigroup(S);;
+gap> MultiplicationTable(S1);
+[ [ 1, 2, 3 ], [ 2, 1, 3 ], [ 3, 3, 3 ] ]
+gap> IsMonoid(S1);
+true
+]]></Example>
+    </Description>
+  </ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="StrongSemilatticeOfSemigroups">
   <ManSection>
     <Oper Name="StrongSemilatticeOfSemigroups" Arg="D, L, H"/>

doc/z-chap08.xml

@@ -37,6 +37,18 @@
   <!--**********************************************************************-->
   <!--**********************************************************************-->
 
+  <Section>
+    <Heading> Power semigroups </Heading>
+    The <E>power semigroup</E> of a semigroup <M>S</M> is the semigroup
+    consisting of the power set of <C>S</C> excluding the empty set, along with
+    the binary operation of setwise multiplication, defined by
+    <Display>(X, Y) \mapsto \{xy : x \in X, y \in Y\}</Display>.
+    <#Include Label = "PowerSemigroup">
+  </Section>
+
+  <!--**********************************************************************-->
+  <!--**********************************************************************-->
+
   <Section Label = "Strong semilattices of semigroups">
     <Heading>
       Strong semilattices of semigroups

gap/semigroups/semicons.gd

@@ -22,6 +22,7 @@ DeclareGlobalFunction("LeftZeroSemigroup");
 DeclareGlobalFunction("RightZeroSemigroup");
 DeclareConstructor("BrandtSemigroupCons", [IsSemigroup, IsGroup, IsPosInt]);
 DeclareGlobalFunction("BrandtSemigroup");
+DeclareOperation("PowerSemigroup", [IsSemigroup and IsFinite]);
 
 DeclareCategory("IsSSSE", IsAssociativeElement);
 DeclareCategoryCollections("IsSSSE");

gap/semigroups/semicons.gi

@@ -967,6 +967,61 @@ for _IsXSemigroup in ["IsTransformationSemigroup",
 od;
 Unbind(_IsXSemigroup);
 
+InstallMethod(PowerSemigroup, "for a semigroup",
+[IsSemigroup and IsFinite],
+function(S)
+  local n, N, subs, x, getindex, multtable, precompute, subsproduct,
+        table, result;
+
+  n := Size(S);
+  if n > 12 then
+    Info(InfoWarning, 1,
+         "PowerSemigroup: the argument has ", n, " elements, so the result ",
+         "will have 2^", n, " - 1 elements and its multiplication table ",
+         "may exceed memory limits, ");
+  fi;
+
+  N := (2 ^ n) - 1;
+
+  # want subsets in binary-code order, not lexicographical, so easier to index
+  # e.g. {1, 2, 3} -> {1}, {2}, {1, 2}, {3}..., i.e. 001, 010, 011, 100...
+  subs := [[]];
+  for x in [1 .. n] do
+    Append(subs, List(subs, s -> Concatenation(s, [x])));
+  od;
+  Remove(subs, 1);
+
+  getindex := s -> Sum(s, j -> 2 ^ (j - 1));
+
+  multtable := MultiplicationTable(S);
+
+  # precompute[i][j] is the product: (element i of S) times (subset j of S)
+  precompute := List([1 .. n],
+                     i -> List(subs, sub -> Set(sub, j -> multtable[i][j])));
+
+  # the product of two subsets X, Y is equivalent to
+  # (x_1 * Y) union (x_2 * Y) union (x_3 * Y)...
+  # so is just a union of the precomputed values above
+  subsproduct := {sub, subindex} -> Union(List(sub,
+                                               i -> precompute[i][subindex]));
+
+  table := List(subs,
+                sub -> List([1 .. N],
+                            subindex -> getindex(subsproduct(sub, subindex))));
+
+  if IsMonoidAsSemigroup(S) then
+    result := MonoidByMultiplicationTable(table);
+  else
+    result := SemigroupByMultiplicationTable(table);
+  fi;
+
+  if IsCommutative(S) then
+    SetIsCommutative(result, true);
+  fi;
+
+  return result;
+end);
+
 InstallMethod(StrongSemilatticeOfSemigroups,
 "for a digraph, a list, and a list",
 [IsDigraph, IsList, IsList],

tst/standard/semigroups/semicons.tst

@@ -972,6 +972,27 @@ gap> S := BrandtSemigroup(IsIntegerMatrixSemigroup, DihedralGroup(4), 3);
 Error, no method found! For debugging hints type ?Recovery from NoMethodFound
 Error, no 1st choice method found for `BrandtSemigroupCons' on 3 arguments
 
+# constructions: power semigroups
+gap> PowerSemigroup(FreeSemigroup(1));
+Error, no method found! For debugging hints type ?Recovery from NoMethodFound
+Error, no 1st choice method found for `PowerSemigroup' on 1 arguments
+gap> S := PowerSemigroup(TrivialSemigroup());;
+gap> IsTrivial(S);
+true
+gap> S := PowerSemigroup(CyclicGroup(2));;
+gap> HasIsCommutative(S);
+true
+gap> IsMonoid(S);
+true
+gap> S1 := SemigroupByMultiplicationTable([[1, 2, 3], [2, 1, 3], [3, 3, 3]]);;
+gap> IsomorphismSemigroups(S, S1) <> fail;
+true
+gap> S := PowerSemigroup(LeftZeroSemigroup(2));;
+gap> HasIsCommutative(S);
+false
+gap> IsMonoid(S);
+false
+
 # constructions: strong semilattices of semigroups: trivial argument checks
 gap> D := CompleteDigraph(2);;
 gap> S1 := TrivialSemigroup();;