Gis trace

gap/congruences/congsemigraph.gd

@@ -26,3 +26,12 @@ DeclareOperation("MinimalHereditarySubsetsVertex",
                  [IsGraphInverseSemigroup, IsPosInt]);
 
 DeclareAttribute("GeneratingCongruencesOfLattice", IsGraphInverseSemigroup);
+
+DeclareCategory("IsTraceOfCongruenceByWangPair",
+    IsSemigroupCongruence
+    and CanComputeEquivalenceRelationPartition
+    and IsMagmaCongruence
+    and IsAttributeStoringRep
+    and IsFinite);
+
+DeclareAttribute("TraceOfCongruenceByWangPair", IsCongruenceByWangPair);

gap/congruences/congsemigraph.gi

@@ -1,8 +1,9 @@
 ############################################################################
 ##
 ##  congsemigraph.gi
-##  Copyright (C) 2022                     Marina Anagnostopoulou-Merkouri
+##  Copyright (C) 2022-2026                Marina Anagnostopoulou-Merkouri
 ##                                                          James Mitchell
+##                                                             Joseph Ward
 ##
 ##  Licensing information can be found in the README file of this package.
 ##
@@ -287,3 +288,104 @@ InstallMethod(TrivialCongruence,
 "for a graph inverse semigroup",
 [IsGraphInverseSemigroup],
 S -> AsCongruenceByWangPair(SemigroupCongruence(S, [])));
+
+InstallMethod(TraceOfCongruenceByWangPair,
+"for a congruence by Wang pair",
+[IsCongruenceByWangPair],
+function(cong)
+  local S, fam, tr;
+
+  S := IdempotentGeneratedSubsemigroup(Source(cong));
+
+  fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(S)),
+                               ElementsFamily(FamilyObj(S)));
+  tr := Objectify(NewType(fam, IsTraceOfCongruenceByWangPair),
+                  rec(cong := cong));
+  SetSource(tr, S);
+  SetRange(tr, S);
+  return tr;
+end);
+
+InstallMethod(ViewObj, "for trace of a congruence by Wang pair",
+[IsTraceOfCongruenceByWangPair],
+function(tr)
+  Print(ViewString(tr));
+end);
+
+InstallMethod(ViewString, "for a congruence by Wang pair",
+[IsTraceOfCongruenceByWangPair],
+function(tr)
+  return StringFormatted(
+    "<trace of graph inverse semigroup congruence with H = {} and W = {}>",
+    ViewString(tr!.cong!.H),
+    ViewString(tr!.cong!.W));
+end);
+
+# TODO write an explanation of what the next function implements
+# TODO add more examples
+InstallMethod(CongruenceTestMembershipNC,
+"for the trace of a congruence by Wang pair",
+[IsTraceOfCongruenceByWangPair,
+ IsGraphInverseSemigroupElement,
+ IsGraphInverseSemigroupElement],
+function(tr, elm1, elm2)
+  local p1, p2, range_elm1_in_H, range_elm2_in_H, tmp, S, e, i;
+
+  if elm1 = elm2 then
+    return true;
+  fi;
+
+  p1 := PositivePath(elm1);
+  p2 := PositivePath(elm2);
+
+  range_elm1_in_H := IsMultiplicativeZero(Source(tr), elm1)
+    or IndexOfVertexOfGraphInverseSemigroup(Range(p1)) in tr!.cong!.H;
+  range_elm2_in_H := IsMultiplicativeZero(Source(tr), elm2)
+    or IndexOfVertexOfGraphInverseSemigroup(Range(p2)) in tr!.cong!.H;
+
+  if range_elm1_in_H or range_elm2_in_H then
+    return range_elm1_in_H and range_elm2_in_H;
+  elif Source(elm1) <> Source(elm2) or Range(elm1) <> Range(elm2) then
+    return false;
+  fi;
+  if Length(p1![1]) > Length(p2![1]) then
+    tmp := p1;
+    p1 := p2;
+    p2 := tmp;
+  fi;
+
+  for i in [1 .. Length(p1![1])] do
+    if p1![1][i] <> p2![1][i] then
+      return false;
+    fi;
+  od;
+
+  S := Source(tr!.cong);
+  for i in [Length(p1![1]) .. Length(p2![1])] do
+    e := EdgesOfGraphInverseSemigroup(S)[p2![1][i]];
+    if not IndexOfVertexOfGraphInverseSemigroup(Source(e)) in tr!.cong!.W then
+      return false;
+    fi;
+  od;
+
+  return true;
+end);
+
+# TODO add family check to install method below
+InstallMethod(ImagesElm,
+"for the trace of a congruence by Wang pair and an element",
+[IsTraceOfCongruenceByWangPair, IsGraphInverseSemigroupElement],
+function(tr, x)
+
+    if not IsIdempotent(x) then
+      Error("TODO");
+    elif not x in Source(tr) then
+      # TODO if family check done above then remove this clause.
+      Error("TODO");
+    fi;
+
+    #TODO
+
+
+end);
+

gap/semigroups/semigraph.gd

@@ -27,6 +27,10 @@ DeclareAttribute("Source", IsGraphInverseSemigroupElement);
 
 DeclareOperation("IsVertex", [IsGraphInverseSemigroupElement]);
 
+DeclareAttribute("PositivePath", IsGraphInverseSemigroupElement);
+DeclareAttribute("NegativePath", IsGraphInverseSemigroupElement);
+# TODO IsEdge?
+
 InstallTrueMethod(IsGeneratorsOfInverseSemigroup,
                   IsGraphInverseSemigroupElementCollection);
 

gap/semigroups/semigraph.gi

@@ -8,6 +8,10 @@
 #############################################################################
 ##
 
+# TODO: Implement special methods for:
+# * NrIdempotents
+# * IdempotentGeneratedSubsemigroup
+
 InstallMethod(AsMonoid, "for a graph inverse semigroup",
 [IsGraphInverseSemigroup], ReturnFail);
 
@@ -302,3 +306,25 @@ InstallMethod(IsWholeFamily,
 "for a subsemigroup of a graph inverse semigroup",
 [IsGraphInverseSubsemigroup],
 S -> Size(ElementsFamily(FamilyObj(S))!.semigroup) = Size(S));
+
+InstallMethod(PositivePath,
+"for a graph inverse semigroup element",
+[IsGraphInverseSemigroupElement],
+function(elt)
+  local pos, S;
+
+  pos := PositionProperty(elt![1], IsNegInt);
+  if pos = fail then
+    return elt;
+  elif pos = 1 then
+    return Source(elt);
+  fi;
+  # Get the semigroup containing "elt"
+  S := FamilyObj(elt)!.semigroup;
+  return EvaluateWord(GeneratorsOfSemigroup(S), elt![1]{[1 .. pos - 1]});
+end);
+
+InstallMethod(NegativePath,
+"for a graph inverse semigroup element",
+[IsGraphInverseSemigroupElement],
+elt -> PositivePath(elt ^ -1) ^ -1);

tst/standard/semigroups/semigraph.tst

@@ -148,6 +148,26 @@ false
 gap> G.1 = H.1;
 false
 
+# Positive/NegativePath
+gap> D := ChainDigraph(4);
+<immutable chain digraph with 4 vertices>
+gap> S := GraphInverseSemigroup(D);
+<finite graph inverse semigroup with 4 vertices, 3 edges>
+gap> x := S.2 * S.3 * S.3 ^ -1 * S.2 ^ -1 * S.1 ^ -1;
+e_2e_3e_3^-1e_2^-1e_1^-1
+gap> NegativePath(x);
+e_3^-1e_2^-1e_1^-1
+gap> x = PositivePath(x) * NegativePath(x);
+true
+gap> Size(S);
+31
+gap> ForAll(S, x -> x = PositivePath(x) * NegativePath(x));
+true
+gap> PositivePath(MultiplicativeZero(S));
+0
+gap> NegativePath(MultiplicativeZero(S));
+0
+
 #
 gap> SEMIGROUPS.StopTest();
 gap> STOP_TEST("Semigroups package: standard/semigroups/semigraph.tst");