Fix dual subsemigroups

semigroups · Mar 29, 2018 · 3d85769 · 3d85769
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diff --git a/doc/dual.xml b/doc/dual.xml
@@ -1,7 +1,7 @@
 #############################################################################
 ##
 #W dual.xml
-#Y Copyright (C) 2011-14 Finn Smith 
+#Y Copyright (C) 2018  Finn Smith
 ##
 ## Licensing information can be found in the README file of this package.
 ##
@@ -18,18 +18,19 @@
  antisomorphic semigroup with the same underlying set as <A>S</A>,
  where multiplication is reversed.
  Note that the dual <A>D</A> of a semigroup <A>S</A> created using 
- <Ref Func="DualSemigroup"> has the opposite value of 
- <Ref Filt="IsDualSemigroup"> to <A>S</A>.
+ <Ref Func="DualSemigroup"/> has the opposite value of
+ <Ref Filt="IsDualSemigroup"/> to <A>S</A>.
 <Example>
 <![CDATA[
-gap> S := Semigroup([Transformation([1, 4, 3, 2, 2]), Transformation([5, 4, 4, 1, 2])]);;
+gap> S := Semigroup([Transformation([1, 4, 3, 2, 2]),
+> Transformation([5, 4, 4, 1, 2])]);;
 gap> D := DualSemigroup(S);
-<dual semigroup of <transformation semigroup of degree 5 with 2 generators>>
+<dual semigroup of <transformation semigroup of degree 5 with 2 
+ generators>>
 gap> Size(S) = Size(D);
 true
 gap> NrDClasses(S) = NrDClasses(D);
-true
-]]> </Example> </Description> </ManSection>
+true]]></Example> </Description> </ManSection>
 <#/GAPDoc>
 
 <#GAPDoc Label="DualSemigroupElement">
@@ -41,25 +42,29 @@ true
  dual semigroup of <A>sgrp</A>. 
 <Example>
 <![CDATA[
-gap> S := Semigroup([Transformation([1, 4, 3, 2, 2]), Transformation([5, 4, 4, 1, 2])]);;
+gap> S := Semigroup([Transformation([1, 4, 3, 2, 2]),
+> Transformation([5, 4, 4, 1, 2])]);;
 gap> D := DualSemigroup(S);
-<dual semigroup of <transformation semigroup of degree 5 with 2 generators>>
-gap> x := Representative(DClasses(S)[1]);; y := Representative(DClasses(S)[2]);;
-gap> dx := DualSemigroupElement(D, x);; dy := DualSemigroupElement(D, y);;
+<dual semigroup of <transformation semigroup of degree 5 with 2 
+ generators>>
+gap> x := Representative(DClasses(S)[1]);;
+gap> y := Representative(DClasses(S)[2]);;
+gap> dx := DualSemigroupElement(D, x);;
+gap> dy := DualSemigroupElement(D, y);;
 gap> dx * dy = DualSemigroupElement(D, y * x);
 true
 gap> dx * dy = DualSemigroupElement(D, x * y);
-false
-]]> </Example> </Description> </ManSection>
+false]]></Example> </Description> </ManSection>
 <#/GAPDoc>
 
 <#GAPDoc Label="IsDualSemigroupElement">
  <ManSection>
  <Filt Name = "IsDualSemigroupElement" Type = "Category" Arg="elt"/>
- <Returns>Returns true if <A>elt</A> has the representation of a dual semigroup element.</Returns>
+ <Returns>Returns true if <A>elt</A> has the representation of a dual
+ semigroup element.</Returns>
  <Description>
  Elements of a dual semigroup created using
- <Ref Func = "DualSemigroupElement"> normally lie in this
+ <Ref Func = "DualSemigroupElement"/> normally lie in this
  category. The exception is dual elements to elements that already
  lie in the category.
 <Example>
@@ -75,8 +80,7 @@ true
 gap> x := DualSemigroupElement(S, t);
 <block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ], [ 4, -4 ]>
 gap> IsDualSemigroupElement(x);
-false
-]]> </Example> </Description> </ManSection>
+false]]></Example> </Description> </ManSection>
 <#/GAPDoc>
 
 <#GAPDoc Label="IsDualSemigroup">
@@ -85,7 +89,7 @@ false
  <Returns>Returns true if <A>sgrp</A> is represented as
  a dual semigroup.</Returns>
  <Description>
- Semigroups created using <Ref Func="DualSemigroup">
+ Semigroups created using <Ref Func="DualSemigroup"/>
  normally lie in this category. The exception is dual
  semigroups to semigroups that already lie in the category.
 <Example>
@@ -94,7 +98,8 @@ gap> S := Semigroup([Transformation([3, 5, 1, 1, 2]),
 > Transformation([1, 2, 4, 4, 3])]);
 <transformation semigroup of degree 5 with 2 generators>
 gap> D := DualSemigroup(S);
-<dual semigroup of <transformation semigroup of degree 5 with 2 generators>>
+<dual semigroup of <transformation semigroup of degree 5 with 2 
+ generators>>
 gap> IsDualSemigroup(D);
 true
 gap> R := DualSemigroup(D);
@@ -106,6 +111,5 @@ true
 gap> T := Range(IsomorphismTransformationSemigroup(D));
 <transformation semigroup of size 16, degree 17 with 2 generators>
 gap> IsDualSemigroup(T);
-false
-]]> </Example> </Description> </ManSection>
+false]]></Example> </Description> </ManSection>
 <#/GAPDoc>
diff --git a/gap/attributes/dual.gd b/gap/attributes/dual.gd
@@ -17,8 +17,10 @@ DeclareSynonym("IsDualSemigroup", IsSemigroup and
 DeclareAttribute("TypeDualSemigroupElements", IsDualSemigroup);
 
 DeclareAttribute("DualSemigroupOfFamily", IsFamily);
+DeclareAttribute("ParentAttr", IsDualSemigroup);
 
 DeclareGlobalFunction("DualSemigroupElement");
 DeclareGlobalFunction("DualSemigroupElementNC");
+DeclareGlobalFunction("UnderlyingElementOfDualSemigroupElement");
 
 DeclareAttribute("AntiIsomorphismDualSemigroup", IsSemigroup);
diff --git a/gap/attributes/dual.gi b/gap/attributes/dual.gi
@@ -14,6 +14,14 @@ InstallMethod(DualSemigroup, "for a semigroup",
 function(S)
  local dual, fam, filts, type;
 
+ if IsDualSemigroup(S) then
+ if HasGeneratorsOfSemigroup(S) then
+ return Semigroup(List(GeneratorsOfSemigroup(S),
+ x -> UnderlyingElementOfDualSemigroupElement(x)));
+ fi;
+ # FS: need to work out what to do if not
+ fi;
+
  fam := NewFamily("DualSemigroupElementsFamily", IsDualSemigroupElement);
  dual := Objectify(NewType(CollectionsFamily(fam),
  IsWholeFamily and
@@ -101,6 +109,17 @@ function(S)
  return MappingByFunction(S, dual, iso, inv);
 end);
 
+InstallGlobalFunction(UnderlyingElementOfDualSemigroupElement,
+"for a dual semigroup element",
+function(s)
+ if not IsDualSemigroupElement(s) then
+ ErrorNoReturn("Semigroups: UnderlyingElementOfDualSemigroupElement: \n",
+ "the argument must be an element represented as a dual ",
+ "semigroup element,");
+ fi;
+ return s![1];
+end);
+
 ################################################################################
 ## Technical methods
 ################################################################################
@@ -130,6 +149,9 @@ end);
 InstallMethod(Representative, "for a dual semigroup",
 [IsDualSemigroup],
 function(S)
+ if HasGeneratorsOfSemigroup(S) then
+ return GeneratorsOfSemigroup(S)[1];
+ fi;
  return DualSemigroupElementNC(S, Representative(DualSemigroup(S)));
 end);
 
@@ -198,3 +220,16 @@ function(x, data)
  end;
  return rec(func := hashfunc, data := H.data);
 end);
+
+InstallMethod(ParentAttr, "for a subsemigroup of a dual semigroup",
+[IsDualSemigroup],
+function(S)
+ return DualSemigroupOfFamily(FamilyObj(Representative(S)));
+end);
+
+InstallMethod(TypeDualSemigroupElements,
+"for a subsemigroup of a dual semigroup",
+[IsDualSemigroup],
+function(S)
+ return TypeDualSemigroupElements(ParentAttr(S));
+end);
diff --git a/gap/tools/utils.gi b/gap/tools/utils.gi
@@ -242,6 +242,7 @@ SEMIGROUPS.DocXMLFiles := ["../PackageInfo.g",
  "congrees.xml",
  "congrms.xml",
  "conguniv.xml",
+ "dual.xml",
  "display.xml",
  "elements.xml",
  "factor.xml",

diff --git a/read.g b/read.g
@@ -81,7 +81,6 @@ ReadPackage("semigroups", "gap/attributes/maximal.gi");
 ReadPackage("semigroups", "gap/attributes/normalizer.gi");
 ReadPackage("semigroups", "gap/attributes/properties.gi");
 
-
 ReadPackage("semigroups", "gap/congruences/congpairs.gi");
 ReadPackage("semigroups", "gap/congruences/congrms.gi");
 ReadPackage("semigroups", "gap/congruences/conguniv.gi");

diff --git a/tst/standard/dual.tst b/tst/standard/dual.tst
@@ -98,7 +98,7 @@ gap> S := Semigroup([Transformation([2, 6, 3, 2, 4, 2]),
 gap> T := DualSemigroup(S);;
 gap> ForAll(DClasses(T),
 > x -> AsSortedList(List(x, y -> DualSemigroupElement(S, y))) =
-> AsSortedList(GreensDClassOfElement(S, 
+> AsSortedList(GreensDClassOfElement(S,
 > DualSemigroupElement(S, Representative(x)))));
 true
 gap> ForAll(DClasses(T),
@@ -121,8 +121,7 @@ true
 gap> S := FullTransformationMonoid(20);;
 gap> T := DualSemigroup(S);;
 gap> Representative(T);
-<Transformation( [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
- 19, 20, 1 ] ) in the dual semigroup>
+<IdentityTransformation in the dual semigroup>
 
 #T# Size
 gap> S := FullTransformationMonoid(20);;
@@ -179,12 +178,100 @@ x ) ... end )
 gap> ForAll(S, x -> (x ^ antiso) ^ invantiso = x);
 true
 
+#T# Subsemigroups of a dual semigroup
+gap> S := FullBooleanMatMonoid(4);
+<monoid of 4x4 boolean matrices with 7 generators>
+gap> T := DualSemigroup(S);
+<dual semigroup of <monoid of 4x4 boolean matrices with 7 generators>>
+gap> U := Semigroup(GeneratorsOfSemigroup(T){[3 .. 5]});
+<dual semigroup of <semigroup of 4x4 boolean matrices with 3 generators>>
+gap> AsList(U);
+[ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 1, 0], 
+ [0, 0, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], 
+ [0, 0, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], 
+ [1, 0, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0], 
+ [0, 0, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 0, 1], [1, 1, 1, 1], 
+ [0, 0, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 1, 0], 
+ [1, 0, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 0, 1, 1], 
+ [0, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], 
+ [0, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [1, 1, 0, 1], 
+ [1, 0, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 1, 0], 
+ [1, 1, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], 
+ [0, 0, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0], 
+ [1, 0, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 1], [1, 1, 1, 1], 
+ [0, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], 
+ [0, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 0, 1], [1, 1, 1, 1], 
+ [1, 0, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0], 
+ [1, 1, 0, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 0, 1], [1, 1, 1, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 0, 1, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 1], [1, 1, 1, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 0], [1, 1, 1, 0], [1, 1, 1, 0], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 0, 0], [1, 0, 1, 0], [1, 1, 0, 1], 
+ [1, 1, 1, 1]]) in the dual semigroup>, 
+ <Matrix(IsBooleanMat, [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1], 
+ [1, 0, 1, 1]]) in the dual semigroup> ]
+gap> Size(last);
+26
+gap> Size(DualSemigroup(U));
+26
+gap> V := DualSemigroup(U);
+<semigroup of size 26, 4x4 boolean matrices with 3 generators>
+gap> V = Semigroup(GeneratorsOfSemigroup(S){[3 .. 5]});
+true
+
+#T# UnderlyingElementOfDualSemigroupElement
+gap> S := SingularPartitionMonoid(4);;
+gap> D := DualSemigroup(S);;
+gap> d := Representative(D);
+<<block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ], [ 4, -4 ]>
+ in the dual semigroup>
+gap> UnderlyingElementOfDualSemigroupElement(d);
+<block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ], [ 4, -4 ]>
+gap> Representative(S);
+<block bijection: [ 1, 2, -1, -2 ], [ 3, -3 ], [ 4, -4 ]>
+gap> UnderlyingElementOfDualSemigroupElement(S);
+Error, Semigroups: UnderlyingElementOfDualSemigroupElement: 
+the argument must be an element represented as a dual semigroup element,
+gap> T := Semigroup(GeneratorsOfSemigroup(D){[1000 .. 2000]});
+<dual semigroup of <bipartition semigroup of degree 4 with 1001 generators>>
+gap> UnderlyingElementOfDualSemigroupElement(Representative(T));
+<bipartition: [ 1, 2, -1 ], [ 3, -2 ], [ 4 ], [ -3, -4 ]>
+
 #T# UnbindVariables
 gap> Unbind(antiso);
 gap> Unbind(inv);
 gap> Unbind(invantiso);
 gap> Unbind(i);
 gap> Unbind(U);
+gap> Unbind(V);
 gap> Unbind(S);
 gap> Unbind(T);