Implemented essential dependency check. (#26)

* Added function to find permutation groups.

* Attempt to add idempotents no 1.

* Co-authored-by: Thomas Gao <thomasgaozx@users.noreply.github.com>

* Extracting idempotents as transformations.

* Almost done essential dependency.

* It works now.

* cleanup.

* more cleanup.

* Removed extra file.

* Added headers and test.

---------

Co-authored-by: Thomas Gao <z72gao@edu.uwaterloo.ca>
Co-authored-by: Srijan01 <khatrisrijan99@gmail.com>

init.g

@@ -14,3 +14,4 @@ ReadPackage("SgpDec","/lib/cascadegroup.gd");
 ReadPackage("SgpDec","/lib/fl.gd");
 ReadPackage("SgpDec","/lib/coords.gd");
 ReadPackage("SgpDec","/lib/holonomy.gd");
+ReadPackage("SgpDec","/lib/lowerbound.gd");

lib/lowerbound.gd

@@ -0,0 +1,13 @@
+#############################################################################
+##
+## lowerbound.gd           SgpDec package
+##
+## Copyright (C) 2023
+##
+## Thomas Gao, Chrystopher L. Nehaniv
+##
+## Checks essential dependency for Rhodes–Tilson complexity lower bound.
+##
+
+DeclareGlobalFunction("IdempotentsForSubset");
+DeclareGlobalFunction("CheckEssentialDependency");

lib/lowerbound.gi

@@ -0,0 +1,67 @@
+#############################################################################
+##
+## lowerbound.gd           SgpDec package
+##
+## Copyright (C) 2023
+##
+## Thomas Gao, Chrystopher L. Nehaniv
+##
+## Checks essential dependency for Rhodes–Tilson complexity lower bound.
+##
+
+InstallGlobalFunction(IdempotentsForSubset,
+function(S, sk, set)
+local e, IdempotentSet, ssize;
+  ssize := Size(BaseSet(sk));
+  IdempotentSet := [];
+  for e in Idempotents(S) do
+    # check whether the image of e is set
+    if OnFiniteSet(BaseSet(sk), e) = set then
+       Add(IdempotentSet, e);
+    fi;
+  od;
+  return IdempotentSet;
+end);
+
+InstallGlobalFunction(CheckEssentialDependency, function(S, sk, d1, d2)
+  # d1 is lower than d2 so larger set
+  local G1, CalX2, CalI2, skJ, J, x1, x2, e1, Xt, JGroup;
+
+  for x1 in Concatenation(SubductionClassesOnDepth(sk, d1)) do
+    for x2 in Concatenation(SubductionClassesOnDepth(sk, d2)) do
+      if IsSubsetBlist(x1, x2) then
+
+        for e1 in IdempotentsForSubset(S, sk, x1) do
+          G1 := SchutzenbergerGroup(HClass(S, e1));
+
+          CalX2 := Enumerate(Orb(G1, x2, OnFiniteSet, rec(schreier := true, orbitgraph := true)));
+          CalI2 := []; #Cal_I2 = all idempotents with images that are members of CalX2;
+          for Xt in CalX2 do
+            CalI2 := Concatenation(CalI2, IdempotentsForSubset(S, sk, Xt));
+          od;
+
+          if IsEmpty(CalI2) then
+            continue;
+          fi;
+
+          J := Semigroup(CalI2);
+          skJ := Skeleton(J);
+
+          if ContainsSet( skJ, x2 ) then
+            JGroup := PermutatorGroup(skJ, x2);
+
+            if Size(JGroup) > 1 then # the stricter test is PermutatorGroup(sk, x2) = JGroup
+              Assert( 1, PermutatorGroup(sk, x2) = JGroup, 
+                Concatenation("PermutatorGroup(sk, x2) <> JGroup\nx2 = ", 
+                  TrueValuePositionsBlistString(x2), "\nx1 = ",
+                  TrueValuePositionsBlistString(x1), "\n") );
+              return JGroup;
+            fi;
+          fi;
+        od;
+
+      fi;
+    od;
+  od;
+  return Group(());
+end);

read.g

@@ -16,5 +16,6 @@ ReadPackage("SgpDec","/lib/cascadesemigroup.gi");
 ReadPackage("SgpDec","/lib/cascadegroup.gi");
 ReadPackage("SgpDec","/lib/fl.gi");
 ReadPackage("SgpDec","/lib/holonomy.gi");
+ReadPackage("SgpDec","/lib/lowerbound.gi");
 #just the test calling functions
 ReadPackage("SgpDec","/tst/testfunctions.g");

tst/lowerbound.tst

@@ -0,0 +1,14 @@
+# testing lowerbound
+gap> START_TEST("Sgpdec package: lowerbound.tst");
+gap> LoadPackage("sgpdec", false);;
+gap> SgpDecFiniteSetDisplayOn();;
+gap> S := FullTransformationSemigroup(4);;
+gap> sk := Skeleton(S);;
+gap> CheckEssentialDependency(S, sk, 1,2);
+Group([ (2,3), (1,2), (1,3) ])
+gap> CheckEssentialDependency(S, sk, 2,3);
+Group([ (1,2) ])
+gap> SgpDecFiniteSetDisplayOff();;
+
+#
+gap> STOP_TEST( "Sgpdec package: lowerbound.tst", 10000);

tst/testfunctions.g

@@ -1,6 +1,7 @@
 SgpDecTestInstall := function()
 local test;
   for test in [
+          "lowerbound",
           "disjointuniongroup",
           "WeakControlWords",
           "wreath",