Add IsModularLatticeDigraph (#629)

* Add IsModularLatticeDigraph

* Add IsModularLatticeDigraph

* Add documentation and tests

---------

Co-authored-by: Murray Whyte <mw231@st-andrews.ac.uk>

Author: james-d-mitchell

doc/prop.xml

@@ -1334,6 +1334,40 @@ false]]></Example>
 </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="IsModularLatticeDigraph">
+<ManSection>
+  <Prop Name="IsModularLatticeDigraph" Arg="digraph"/>
+  <Returns><K>true</K> or <K>false</K>.</Returns>
+  <Description>
+    <C>IsModularLatticeDigraph</C> returns <K>true</K> if the digraph
+    <A>digraph</A> is a <E>modular lattice digraph</E>.<P/>
+
+    A <E>modular lattice digraph</E> is a <E>lattice digraph</E>
+    (<Ref Prop="IsLatticeDigraph"/>) which is modular. That is to say,
+    the <E>lattice digraph</E> representing <M>N5</M> is not
+    embeddable as a lattice
+    (see <URL>https://en.wikipedia.org/wiki/Modular_lattice</URL> and
+    <Ref Prop="IsLatticeEmbedding"/>). 
+
+    &MUTABLE_RECOMPUTED_PROP;
+
+    <Example><![CDATA[
+gap> D := ChainDigraph(5);
+<immutable chain digraph with 5 vertices>
+gap> D := DigraphReflexiveTransitiveClosure(D);
+<immutable preorder digraph with 5 vertices, 15 edges>
+gap> IsModularLatticeDigraph(D);
+true
+gap> N5 := Digraph([[2, 3], [4], [4], []]);
+<immutable digraph with 4 vertices, 4 edges>
+gap> DigraphReflexiveTransitiveClosure(N5);
+<immutable preorder digraph with 4 vertices, 9 edges>
+gap> IsModularLatticeDigraph(N5);
+false
+]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
 
 <#GAPDoc Label="IsPreorderDigraph">
 <ManSection>

doc/z-chap5.xml

@@ -36,6 +36,7 @@
     <#Include Label="DigraphMeetTable">
     <#Include Label="IsUpperSemimodularDigraph">
     <#Include Label="IsDistributiveLatticeDigraph">
+    <#Include Label="IsModularLatticeDigraph">
   </Section>
 
   <Section><Heading>Regularity</Heading>

gap/prop.gd

@@ -51,6 +51,7 @@ DeclareProperty("IsJoinSemilatticeDigraph", IsDigraph);
 DeclareProperty("IsMeetSemilatticeDigraph", IsDigraph);
 DeclareProperty("IsPermutationDigraph", IsDigraph);
 DeclareProperty("IsDistributiveLatticeDigraph", IsDigraph);
+DeclareProperty("IsModularLatticeDigraph", IsDigraph);
 DeclareSynonymAttr("IsLatticeDigraph",
                    IsMeetSemilatticeDigraph and IsJoinSemilatticeDigraph);
 DeclareSynonymAttr("IsPreorderDigraph",

gap/prop.gi

@@ -661,19 +661,28 @@ end);
 
 InstallMethod(IsDistributiveLatticeDigraph, "for a digraph", [IsDigraph],
 function(D)
-  local N5, M3;
+  local M3;
+
   if not IsLatticeDigraph(D) then
     return false;
   fi;
 
-  N5 := DigraphReflexiveTransitiveClosure(
-        Digraph([[2, 4], [3], [5], [5], []]));
   M3 := DigraphReflexiveTransitiveClosure(
         Digraph([[2, 3, 4], [5], [5], [5], []]));
 
-  if LatticeDigraphEmbedding(N5, D) <> fail or
-       LatticeDigraphEmbedding(M3, D) <> fail then
+  return IsModularLatticeDigraph(D) and
+       LatticeDigraphEmbedding(M3, D) = fail;
+end);
+
+InstallMethod(IsModularLatticeDigraph, "for a digraph", [IsDigraph],
+function(D)
+  local N5;
+  if not IsLatticeDigraph(D) then
     return false;
   fi;
-  return true;
+
+  N5 := DigraphReflexiveTransitiveClosure(
+        Digraph([[2, 4], [3], [5], [5], []]));
+
+  return LatticeDigraphEmbedding(N5, D) = fail;
 end);

tst/standard/prop.tst

@@ -1330,6 +1330,33 @@ gap> D := DigraphReflexiveTransitiveClosure(D);
 gap> IsDistributiveLatticeDigraph(D);
 true
 
+# IsModularLatticeDigraph
+gap>  D := Digraph([[11, 10], [], [2], [2], [3], [4, 3], [6, 5], [7], [7],
+>      [8], [9, 8]]);
+<immutable digraph with 11 vertices, 14 edges>
+gap> D := DigraphReflexiveTransitiveClosure(D);
+<immutable preorder digraph with 11 vertices, 60 edges>
+gap> IsModularLatticeDigraph(D);
+true
+gap> D := ChainDigraph(10);
+<immutable chain digraph with 10 vertices>
+gap> D := DigraphReflexiveTransitiveClosure(D);
+<immutable preorder digraph with 10 vertices, 55 edges>
+gap> IsModularLatticeDigraph(D);
+true
+gap> D := Digraph([[2, 4], [3, 5], [9], [5, 6], [7], [7, 8], [9], [9], []]);
+<immutable digraph with 9 vertices, 12 edges>
+gap> D := DigraphReflexiveTransitiveClosure(D);
+<immutable preorder digraph with 9 vertices, 34 edges>
+gap> IsModularLatticeDigraph(D);
+false
+gap> M3 := Digraph([[2, 3, 4], [5], [5], [5], []]);
+<immutable digraph with 5 vertices, 6 edges>
+gap> M3 := DigraphReflexiveTransitiveClosure(M3);
+<immutable preorder digraph with 5 vertices, 12 edges>
+gap> IsModularLatticeDigraph(M3);
+true
+
 # IsPartialOrderDigraph
 gap> gr := NullDigraph(5);
 <immutable empty digraph with 5 vertices>