Cayley digraphs

PackageInfo.g

@@ -11,7 +11,7 @@
 ##  <#GAPDoc Label="PKGVERSIONDATA">
 ##  <!ENTITY VERSION "3.0.4">
 ##  <!ENTITY GAPVERS "4.9.0">
-##  <!ENTITY DIGRAPHSVERS "0.7.1">
+##  <!ENTITY DIGRAPHSVERS "0.10.0">
 ##  <!ENTITY ORBVERS "4.7.5">
 ##  <!ENTITY IOVERS "4.4.4">
 ##  <!ENTITY GENSSVERS "1.5">
@@ -236,7 +236,7 @@ Dependencies := rec(
   GAP := ">=4.9.0",
   NeededOtherPackages := [["orb", ">=4.7.5"],
                           ["io", ">=4.4.4"],
-                          ["digraphs", ">=0.7.1"],
+                          ["digraphs", ">=0.10.0"],
                           ["genss", ">=1.5"]],
   SuggestedOtherPackages := [["gapdoc", ">=1.5.1"]],
 

README.md

@@ -27,7 +27,7 @@ The  following  is  a  summary of the steps that should lead to a successful ins
 * get the [Orb](http://gap-system.github.io/orb/) package version 4.7.5 or higher. 
   Both [Orb](http://gap-system.github.io/orb/) and [Semigroups](https://gap-packages.github.io/Semigroups) perform better if [Orb](http://gap-system.github.io/orb/) is compiled, so compile [Orb](http://gap-system.github.io/orb/)!
 
-* ensure that the [Digraphs](http://gap-system.github.io/digraphs/) package version 0.7.1 or higher is available.  [Digraphs](http://gap-system.github.io/digraphs/) must be compiled before [Semigroups](https://gap-packages.github.io/Semigroups) can be
+* ensure that the [Digraphs](http://gap-system.github.io/digraphs/) package version 0.10.0 or higher is available.  [Digraphs](http://gap-system.github.io/digraphs/) must be compiled before [Semigroups](https://gap-packages.github.io/Semigroups) can be
 loaded.
 
 * get the [genss](http://gap-system.github.io/genss/) package version 1.5 or higher 

doc/display.xml

@@ -127,8 +127,23 @@
         </Item>
         <Mark>pbrs</Mark>
         <Item>
-          If <A>obj</A> is a PBR, then <C>TikzString</C> returns a graphical
-          representation <A>obj</A>; see Section <Ref Sect = "section-blocks"/>.
+          If <A>obj</A> is a <Ref Oper="PBR"/>, then <C>TikzString</C> returns a
+          graphical representation <A>obj</A>; see Chapter <Ref Sect =
+            "Partitioned binary relations (PBRs)"/>.
+        </Item>
+        <Mark>Cayley graphs</Mark>
+        <Item>
+          If <A>obj</A> is a <Ref Oper="Digraph" BookName="digraphs"/> in the
+          category <Ref Filt="IsCayleyDigraph" BookName="digraphs"/>, then
+          <C>TikzString</C> returns a picture of <A>obj</A>. No attempt is made
+          whatsoever to produce a sensible picture of the digraph <A>obj</A>,
+          in fact, the vertices are all given the same coordinates. Human
+          intervention is required to produce a meaningful picture from the
+          value returned by this method. It is intended to make the task of
+          drawing such a Cayley graph more straightforward by providing
+          everything except the final layout of the graph. Please use <Ref
+            Oper="DotString"/> if you want an automatically laid out diagram of
+          the digraph <A>obj</A>.
         </Item>
       </List>
 
@@ -288,6 +303,29 @@ gap> FileString("t3.dot", DotString(S));
   </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="DotStringDigraph">
+  <ManSection>
+    <Oper Name="DotString" Arg="digraph" Label="for a Cayley digraph"/>
+    <Returns>A string.</Returns>
+    <Description>
+      If <A>digraph</A> is a <Ref Oper="Digraph" BookName="digraphs"/> in the
+      category <Ref Filt="IsCayleyDigraph" BookName="digraphs"/>, then
+      <C>DotString</C> returns a graphical representation of <A>digraph</A>. 
+      The output is in <C>dot</C> format (also known
+      as <C>GraphViz</C>) format. For details about this file format, and
+      information about how to display or edit this format see
+      <URL>http://www.graphviz.org</URL>.
+      <P/>
+
+      The string returned by <C>DotString</C> can be written to a file using
+      the command <Ref Func="FileString" BookName="GAPDoc"/>.<P/>
+
+      See also <Ref Oper="DotLeftCayleyDigraph"/> and <Ref
+        Oper="TikzLeftCayleyDigraph"/>.
+    </Description>
+  </ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="DotSemilatticeOfIdempotents">
   <ManSection>
     <Attr Name="DotSemilatticeOfIdempotents" Arg="S"/>
@@ -341,3 +379,69 @@ d{pmatrix}"]]></Example>
     </Description>
   </ManSection>
 <#/GAPDoc>
+
+<#GAPDoc Label="TikzLeftCayleyDigraph">
+  <ManSection>
+    <Oper Name="TikzLeftCayleyDigraph" Arg="S"/>
+    <Oper Name="TikzRightCayleyDigraph" Arg="S"/>
+    <Returns>A string.</Returns>
+    <Description>
+      If <A>S</A> is a semigroup in the representation 
+      <Ref Filt="IsEnumerableSemigroupRep"/>, then <C>TikzLeftCayleyDigraph</C>
+      is simply short for <C>TikzString(LeftCayleyDigraph(<A>S</A>))</C>.<P/>
+
+      <C>TikzRightCayleyDigraph</C> can be used to produce a tikz string for
+      the right Cayley graph of <A>S</A>.<P/>
+
+      See <Ref Oper="TikzString"/> for more details, and see also 
+      <Ref Oper="DotLeftCayleyDigraph"/>.
+
+      <Log><![CDATA[
+gap> TikzLeftCayleyDigraph(Semigroup(IdentityTransformation));
+"\\begin{tikzpicture}[scale=1, auto, \n  vertex/.style={c\
+ircle, draw, thick, fill=white, minimum size=0.65cm},\n  \
+edge/.style={arrows={-angle 90}, thick},\n  loop/.style={\
+min distance=5mm,looseness=5,arrows={-angle 90},thick}]\n\
+\n  % Vertices . . .\n  \\node [vertex] (a) at (0, 0) {};\
+\n  \\node at (0, 0) {$a$};\n\n  % Edges . . .\n  \\path[\
+->] (a) edge [loop]\n           node {$a$} (a);\n\\end{ti\
+kzpicture}"
+gap> TikzRightCayleyDigraph(Semigroup(IdentityTransformation));
+"\\begin{tikzpicture}[scale=1, auto, \n  vertex/.style={c\
+ircle, draw, thick, fill=white, minimum size=0.65cm},\n  \
+edge/.style={arrows={-angle 90}, thick},\n  loop/.style={\
+min distance=5mm,looseness=5,arrows={-angle 90},thick}]\n\
+\n  % Vertices . . .\n  \\node [vertex] (a) at (0, 0) {};\
+\n  \\node at (0, 0) {$a$};\n\n  % Edges . . .\n  \\path[\
+->] (a) edge [loop]\n           node {$a$} (a);\n\\end{ti\
+kzpicture}"]]></Log>
+    </Description>
+  </ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="DotLeftCayleyDigraph">
+  <ManSection>
+    <Oper Name="DotLeftCayleyDigraph" Arg="S"/>
+    <Oper Name="DotRightCayleyDigraph" Arg="S"/>
+    <Returns>A string.</Returns>
+    <Description>
+      If <A>S</A> is a semigroup in the representation <Ref
+        Filt="IsEnumerableSemigroupRep"/>, then <C>DotLeftCayleyDigraph</C> is
+      simply short for <C>DotString(LeftCayleyDigraph(<A>S</A>))</C>.<P/>
+
+      <C>DotRightCayleyDigraph</C> can be used to produce a dot string for
+      the right Cayley graph of <A>S</A>.<P/>
+
+      See <Ref Oper="DotString"/> for more details, and see also 
+      <Ref Oper="TikzLeftCayleyDigraph"/>.
+
+      <Log><![CDATA[
+gap> DotLeftCayleyDigraph(Semigroup(IdentityTransformation));
+"//dot\ndigraph hgn{\nnode [shape=circle]\n1 [label=\"a\"]\n1 -> 1\n}\
+\n"
+gap> DotRightCayleyDigraph(Semigroup(IdentityTransformation));
+"//dot\ndigraph hgn{\nnode [shape=circle]\n1 [label=\"a\"]\n1 -> 1\n}\
+\n"]]></Log>
+    </Description>
+  </ManSection>
+<#/GAPDoc>

doc/fropin.xml

@@ -50,8 +50,8 @@ false
       returns a list of the elements of <A>S</A> in the order they are
       enumerated by the Froidure-Pin Algorithm. This is the same as the order
       used to index the elements of <A>S</A> in <Ref
-        Attr="RightCayleyGraphSemigroup"/> and <Ref
-        Attr="LeftCayleyGraphSemigroup"/>. <P/>
+        Attr="RightCayleyDigraph"/> and <Ref
+        Attr="LeftCayleyDigraph"/>. <P/>
 
       <C>EnumeratorCanonical</C> and <C>IteratorCanonical</C> return an
       enumerator and an iterator where the elements are
@@ -67,8 +67,8 @@ false
       <C>AsList</C> may not equal the value returned by <C>AsListCanonical</C>.
       <C>AsListCanonical</C> exists so that there is a method for obtaining the
       elements of <A>S</A> in the particular order used by 
-      <Ref Attr="RightCayleyGraphSemigroup"/> and 
-      <Ref Attr="LeftCayleyGraphSemigroup"/>.<P/>
+      <Ref Attr="RightCayleyDigraph"/> and 
+      <Ref Attr="LeftCayleyDigraph"/>.<P/>
 
       See also <Ref Oper="PositionCanonical"/>.
 
@@ -183,27 +183,34 @@ true]]></Example>
   </ManSection>
 <#/GAPDoc>
 
-<#GAPDoc Label="RightCayleyGraphSemigroup">
+<#GAPDoc Label="RightCayleyDigraph">
   <ManSection>
-    <Attr Name="RightCayleyGraphSemigroup" Arg="S"/> 
-    <Attr Name="LeftCayleyGraphSemigroup" Arg="S"/> 
+    <Attr Name="RightCayleyDigraph" Arg="S"/> 
+    <Attr Name="LeftCayleyDigraph" Arg="S"/> 
     <Returns>A list of lists of positive integers.</Returns>
     <Description>
       When the argument <A>S</A> is a semigroup in the representation 
       <Ref Filt="IsEnumerableSemigroupRep"/>, 
-      <C>RightCayleyGraphSemigroup</C> returns the right
-      Cayley graphs of <A>S</A>, as a list <C>graph</C> where 
-      <C>graph[i][j]</C> is equal to 
+      <C>RightCayleyDigraph</C> returns the right
+      Cayley graphs of <A>S</A>, as a <Ref Oper="Digraph" BookName="digraphs"/>
+      <C>digraph</C> where vertex <C>OutNeighbours(digraph)[i][j]</C> is 
       <C>PositionCanonical(<A>S</A>, AsListCanonical(<A>S</A>)[i] *
         GeneratorsOfSemigroup(<A>S</A>)[j])</C>.
-      The list returned by <C>LeftCayleyGraphSemigroup</C> is defined analogously.
+      The digraph returned by <C>LeftCayleyDigraph</C> is defined analogously.<P/>
+
+      The digraph returned by this attribute belongs to the category 
+      <Ref Filt="IsCayleyDigraph" BookName="digraphs"/>, the semigroup <A>S</A>
+      and the generators used to create the digraph can be recovered from the
+      digraph using  <Ref Attr="SemigroupOfCayleyDigraph" BookName="digraphs"/>
+      and <Ref Attr="GeneratorsOfCayleyDigraph" BookName="digraphs"/>.
+
       <Example><![CDATA[
 gap> S := FullTransformationMonoid(2);
 <full transformation monoid of degree 2>
-gap> RightCayleyGraphSemigroup(S);
-[ [ 1, 2, 3 ], [ 2, 1, 3 ], [ 3, 4, 3 ], [ 4, 3, 3 ] ]
-gap> LeftCayleyGraphSemigroup(S);
-[ [ 1, 2, 3 ], [ 2, 1, 4 ], [ 3, 3, 3 ], [ 4, 4, 4 ] ]]]></Example>
+gap> RightCayleyDigraph(S);
+<multidigraph with 4 vertices, 12 edges>
+gap> LeftCayleyDigraph(S);
+<multidigraph with 4 vertices, 12 edges>]]></Example>
     </Description>
   </ManSection>
 <#/GAPDoc>

doc/z-chap13.xml

@@ -27,7 +27,7 @@
     <Heading>
        Cayley graphs
      </Heading>
-     <#Include Label = "RightCayleyGraphSemigroup">
+     <#Include Label = "RightCayleyDigraph">
    </Section>
 
   <!--**********************************************************************-->

doc/z-chap18.xml

@@ -46,7 +46,9 @@
       BookName = "GAPDoc"/> or viewed using <Ref Func = "Splash"/>.
 
     <#Include Label = "DotString"/>
+    <#Include Label = "DotStringDigraph"/>
     <#Include Label = "DotSemilatticeOfIdempotents"/>
+    <#Include Label = "DotLeftCayleyDigraph">
 
   </Section>
 
@@ -81,6 +83,7 @@
       BookName = "GAPDoc"/> or viewed using <Ref Func = "Splash"/>.
 
     <#Include Label = "TikzString">
+    <#Include Label = "TikzLeftCayleyDigraph">
 
   </Section>
 

gap/congruences/conginv.gi

@@ -507,7 +507,7 @@ SEMIGROUPS.KernelTraceClosure := function(S, kernel, traceBlocks, pairstoapply)
   hashlen := SEMIGROUPS.OptionsRec(S).hashlen.L;
   ht := HTCreate([1, 1], rec(forflatplainlists := true,
                              treehashsize := hashlen));
-  right := RightCayleyGraphSemigroup(idsmgp);
+  right := OutNeighbours(RightCayleyDigraph(idsmgp));
   genstoapply := [1 .. Length(right[1])];
 
   #

gap/greens/gree.gi

@@ -391,8 +391,6 @@ function(S)
 
   l  := LeftCayleyGraphSemigroup(S);
   r  := RightCayleyGraphSemigroup(S);
-  # WW: in the future, when l and r are digraphs, gr can be created
-  #     by using DigraphEdgeUnion(l, r)
   gr := Digraph(List([1 .. Length(l)], i -> Concatenation(l[i], r[i])));
   gr := QuotientDigraph(gr, DigraphStronglyConnectedComponents(gr).comps);
   return List(OutNeighbours(gr), Set);

gap/greens/gren.gi

@@ -225,19 +225,19 @@ end);
 InstallMethod(GreensRRelation, "for an enumerable semigroup",
 [IsEnumerableSemigroupRep],
 function(S)
-  local fam, rel;
+  local fam, data, rel;
   if IsActingSemigroup(S) then
     TryNextMethod();
   fi;
   fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(S)),
                                ElementsFamily(FamilyObj(S)));
-
+  data := DigraphStronglyConnectedComponents(RightCayleyDigraph(S));
   rel := Objectify(NewType(fam,
                            IsEquivalenceRelation
                              and IsEquivalenceRelationDefaultRep
                              and IsGreensRRelation
                              and IsEnumerableSemigroupGreensRelationRep),
-                   rec(data := GABOW_SCC(RightCayleyGraphSemigroup(S))));
+                   rec(data := data));
   SetSource(rel, S);
   SetRange(rel, S);
   SetIsLeftSemigroupCongruence(rel, true);
@@ -250,19 +250,19 @@ end);
 InstallMethod(GreensLRelation, "for an enumerable semigroup",
 [IsEnumerableSemigroupRep],
 function(S)
-  local fam, rel;
+  local fam, data, rel;
   if IsActingSemigroup(S) then
     TryNextMethod();
   fi;
   fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(S)),
                                ElementsFamily(FamilyObj(S)));
-
+  data := DigraphStronglyConnectedComponents(LeftCayleyDigraph(S));
   rel := Objectify(NewType(fam,
                            IsEquivalenceRelation
                              and IsEquivalenceRelationDefaultRep
                              and IsGreensLRelation
                              and IsEnumerableSemigroupGreensRelationRep),
-                   rec(data := GABOW_SCC(LeftCayleyGraphSemigroup(S))));
+                   rec(data := data));
   SetSource(rel, S);
   SetRange(rel, S);
   SetIsRightSemigroupCongruence(rel, true);
@@ -514,19 +514,20 @@ InstallMethod(HClassReps, "for a Green's class of an enumerable semigroup",
 [IsGreensClass and IsEnumerableSemigroupGreensClassRep],
 C -> SEMIGROUPS.XClassRepsOfClass(C, GreensHRelation));
 
-## Partial order of D-classes
-# There is duplicate code in here and in maximal D-classes
+# There is duplicate code in here and in maximal D-classes.
+#
+# This cannot be replaced with the method for IsSemigroup and IsFinite since
+# the value of GreensDRelation(S)!.data.comps is not the same as the output of
+# DigraphStronglyConnectedComponents.
 
 InstallMethod(PartialOrderOfDClasses, "for a finite enumerable semigroup",
 [IsEnumerableSemigroupRep and IsFinite],
 function(S)
   local l, r, gr;
-
-  l  := LeftCayleyGraphSemigroup(S);
-  r  := RightCayleyGraphSemigroup(S);
-  gr := Digraph(List([1 .. Length(l)], i -> Concatenation(l[i], r[i])));
+  l  := LeftCayleyDigraph(S);
+  r  := RightCayleyDigraph(S);
+  gr := DigraphEdgeUnion(l, r);
   gr := QuotientDigraph(gr, GreensDRelation(S)!.data.comps);
-
   return List(OutNeighbours(gr), Set);
 end);
 

gap/ideals/idealenum.gi

@@ -53,8 +53,8 @@ SEMIGROUPS.EnumerateIdeal := function(enum, limit, lookfunc)
   indices := enum!.indices;
 
   S := SupersemigroupOfIdeal(UnderlyingCollection(enum));
-  left := LeftCayleyGraphSemigroup(S);
-  right := RightCayleyGraphSemigroup(S);
+  left := OutNeighbours(LeftCayleyDigraph(S));
+  right := OutNeighbours(RightCayleyDigraph(S));
   # FIXME Once the left and right Cayley graphs have been calculated, the
   # entire data structure of S is known and from this it is relatively easy to
   # find the entire data structure for I, so there is no point to what follows,

gap/main/fropin.gd

@@ -48,3 +48,6 @@ DeclareOperation("Enumerate", [IsEnumerableSemigroupRep]);
 DeclareOperation("IsFullyEnumerated", [IsEnumerableSemigroupRep]);
 
 DeclareProperty("IsSemigroupEnumerator", IsEnumeratorByFunctions);
+
+DeclareAttribute("LeftCayleyDigraph", IsEnumerableSemigroupRep);
+DeclareAttribute("RightCayleyDigraph", IsEnumerableSemigroupRep);