Make McAlister triple semigroups acting

doc/semieunit.xml

@@ -118,11 +118,7 @@ gap> T := McAlisterTripleSemigroup(A, x, y);
 gap> S = T;
 false
 gap> IsIsomorphicSemigroup(S, T);
-true
-gap> GeneratorsOfSemigroup(T);
-[ (1, ()), (4, ()), (5, ()), (1, (2,4)(3,5)) ]
-gap> AsSemigroup(IsPartialPermSemigroup, T);
-<inverse partial perm monoid of size 4, rank 4 with 4 generators>]]></Example>
+true]]></Example>
   </Description>
   </ManSection>
 <#/GAPDoc>

gap/main/setup.gi

@@ -47,6 +47,15 @@ function(coll)
   return IsPermGroup(UnderlyingSemigroup(R)) and IsRegularSemigroup(R);
 end);
 
+InstallMethod(IsGeneratorsOfActingSemigroup,
+"for a McAlister triple element collection",
+[IsMcAlisterTripleSemigroupElementCollection],
+function(coll)
+  return
+  IsPermGroup(McAlisterTripleSemigroupGroup(MTSEParent(Representative(coll))));
+    # and McAlisterTripleSemigroupAction(MTSEParent(coll[1])) = OnPoints;
+end);
+
 # FIXME with the below uncommented many tests fail
 ## The HasRows and HasColumns could be removed if it was possible to apply
 ## immediate methods to Rees 0-matrix semigroups
@@ -94,6 +103,12 @@ function(x)
   return NrMovedPoints(x![2]) + 1;
 end);
 
+InstallMethod(ActionDegree, "for a McAlister semigroup element",
+[IsMcAlisterTripleSemigroupElement],
+function(x)
+  return 0;
+end);
+
 InstallMethod(ActionDegree, "for a matrix over finite field object",
 [IsMatrixOverFiniteField], DimensionOfMatrixOverSemiring);
 
@@ -124,6 +139,12 @@ function(coll)
   return DimensionOfMatrixOverSemiring(coll[1]);
 end);
 
+InstallMethod(ActionDegree, "for a McAlister semigroup element collection",
+[IsMcAlisterTripleSemigroupElementCollection],
+function(coll)
+  return MaximumList(List(coll, x -> ActionDegree(x)));
+end);
+
 InstallMethod(ActionDegree, "for a transformation semigroup",
 [IsTransformationSemigroup], DegreeOfTransformationSemigroup);
 
@@ -148,6 +169,12 @@ function(R)
   return 0;
 end);
 
+InstallMethod(ActionDegree, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  return 0;
+end);
+
 InstallMethod(ActionDegree, "for a matrix over finite field semigroup",
 [IsMatrixOverFiniteFieldSemigroup],
 function(S)
@@ -217,6 +244,22 @@ function(R)
   end;
 end);
 
+InstallMethod(ActionRank, "for a McAlister triple semigroup element and int",
+[IsMcAlisterTripleSemigroupElement, IsInt],
+function(f, n)
+  local digraph, id;
+  digraph := McAlisterTripleSemigroupQuotientDigraph(MTSEParent(f));
+  digraph := DigraphReverse(digraph);
+  id      := McAlisterTripleSemigroupComponents(MTSEParent(f)).id;
+  return Position(DigraphTopologicalSort(digraph), id[f[1]]);
+end);
+
+InstallMethod(ActionRank, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  return x -> ActionDegree(S);
+end);
+
 InstallMethod(ActionRank, "for a matrix object and integer",
 [IsMatrixOverFiniteField, IsInt],
 function(x, i)
@@ -243,6 +286,9 @@ InstallMethod(MinActionRank, "for a bipartition semigroup",
 InstallMethod(MinActionRank, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], x -> 0);
 
+InstallMethod(MinActionRank, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], x -> 1);
+
 InstallMethod(MinActionRank, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], x -> 0);
 
@@ -260,6 +306,9 @@ InstallMethod(LambdaOrbOpts, "for a bipartition semigroup",
 InstallMethod(LambdaOrbOpts, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], S -> rec());
 
+InstallMethod(LambdaOrbOpts, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S -> rec());
+
 InstallMethod(LambdaOrbOpts, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], s -> rec());
 
@@ -275,6 +324,9 @@ InstallMethod(RhoOrbOpts, "for a bipartition semigroup",
 InstallMethod(RhoOrbOpts, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], S -> rec());
 
+InstallMethod(RhoOrbOpts, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S -> rec());
+
 InstallMethod(RhoOrbOpts, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], s -> rec());
 
@@ -307,6 +359,24 @@ InstallMethod(LambdaAct, "for a Rees 0-matrix subsemigroup",
   fi;
 end);
 
+InstallMethod(LambdaAct, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  local act, digraph;
+  S := MTSEParent(Representative(S));
+  act     := McAlisterTripleSemigroupAction(S);
+  digraph := McAlisterTripleSemigroupPartialOrder(S);
+  return
+  function(pt, x)
+    if pt = 0 then
+      return act(x[1], x[2] ^ -1);
+    fi;
+    return PartialOrderDigraphJoinOfVertices(digraph,
+                                             act(pt, x[2] ^ -1),
+                                             act(x[1], x[2] ^ -1));
+  end;
+end);
+
 InstallMethod(LambdaAct, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup],
 S -> function(vsp, mat)
@@ -345,6 +415,24 @@ InstallMethod(RhoAct, "for a Rees 0-matrix subsemigroup",
   fi;
 end);
 
+InstallMethod(RhoAct, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  local act, digraph;
+  S := MTSEParent(Representative(S));
+  act     := McAlisterTripleSemigroupAction(S);
+  digraph := McAlisterTripleSemigroupPartialOrder(S);
+  return
+    function(pt, x)
+      if pt = 0 then
+        return x[1];
+      fi;
+      return PartialOrderDigraphJoinOfVertices(digraph,
+                                               act(pt, x[2]),
+                                               x[1]);
+    end;
+end);
+
 InstallMethod(RhoAct, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup],
 function(S)
@@ -370,6 +458,9 @@ end);
 InstallMethod(LambdaOrbSeed, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], S -> -1);
 
+InstallMethod(LambdaOrbSeed, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S -> 0);
+
 InstallMethod(LambdaOrbSeed, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup],
 function(S)
@@ -397,6 +488,9 @@ end);
 InstallMethod(RhoOrbSeed, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], S -> -1);
 
+InstallMethod(RhoOrbSeed, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S -> 0);
+
 InstallMethod(RhoOrbSeed, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], LambdaOrbSeed);
 
@@ -427,6 +521,14 @@ InstallMethod(LambdaFunc, "for a Rees 0-matrix subsemigroup",
   return 0;
 end);
 
+InstallMethod(LambdaFunc, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  local act;
+  act := McAlisterTripleSemigroupAction(MTSEParent(Representative(S)));
+  return x -> act(x[1], x[2] ^ -1);
+end);
+
 InstallMethod(LambdaFunc, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup],
 s -> function(mat)
@@ -454,6 +556,12 @@ InstallMethod(RhoFunc, "for a bipartition semigroup",
 InstallMethod(RhoFunc, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], R -> (x -> x![1]));
 
+InstallMethod(RhoFunc, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  return x -> x[1];
+end);
+
 InstallMethod(RhoFunc, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup],
 function(S)
@@ -464,7 +572,7 @@ function(S)
     end;
 end);
 
-# the function used to calculate the rank of lambda or rho value
+# The function used to calculate the rank of lambda or rho value
 
 InstallMethod(LambdaRank, "for a transformation semigroup",
 [IsTransformationSemigroup], x -> Length);
@@ -487,6 +595,17 @@ function(x)
   fi;
 end);
 
+InstallMethod(LambdaRank, "for a McAlister subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S ->
+function(x)
+  local T;
+  if x = 0 then
+    return 0;
+  fi;
+  T := MTSEParent(Representative(S));
+  return ActionRank(MTSE(T, x, One(McAlisterTripleSemigroupGroup(T))), 0);
+end);
+
 # Why are there row spaces and matrices passed in here?
 InstallMethod(LambdaRank, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], x -> Rank);
@@ -512,6 +631,17 @@ InstallMethod(RhoRank, "for a Rees 0-matrix subsemigroup",
 InstallMethod(RhoRank, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], S -> LambdaRank(S));
 
+InstallMethod(RhoRank, "for a McAlister subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S ->
+function(x)
+  local T;
+  if x = 0 then
+    return 0;
+  fi;
+  T := MTSEParent(Representative(S));
+  return ActionRank(MTSE(T, x, One(McAlisterTripleSemigroupGroup(T))), 0);
+end);
+
 # if g=LambdaInverse(X, f) and X^f=Y, then Y^g=X and g acts on the right
 # like the inverse of f on Y.
 
@@ -523,6 +653,11 @@ InstallMethod(LambdaInverse, "for a partial perm semigroup",
                                  return f ^ -1;
                                end);
 
+InstallMethod(LambdaInverse, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S -> function(x, f)
+                                   return f ^ -1;
+                                 end);
+
 InstallMethod(LambdaInverse, "for a bipartition semigroup",
 [IsBipartitionSemigroup], S -> BLOCKS_INV_RIGHT);
 
@@ -559,6 +694,11 @@ InstallMethod(RhoInverse, "for a partial perm semigroup",
     return f ^ -1;
   end);
 
+InstallMethod(RhoInverse, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S -> function(x, f)
+                                        return f ^ -1;
+                                      end);
+
 # JDM better method for this!!
 
 InstallMethod(RhoInverse, "for a Rees 0-matrix subsemigroup",
@@ -636,6 +776,17 @@ end);
 InstallMethod(RhoBound, "for a Rees 0-matrix semigroup",
 [IsReesZeroMatrixSubsemigroup], LambdaBound);
 
+InstallMethod(LambdaBound, "for a McAlister subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S ->
+function(r)
+  local G;
+  G := McAlisterTripleSemigroupGroup(MTSEParent(Representative(S)));
+  return Size(Stabilizer(G, r));
+end);
+
+InstallMethod(RhoBound, "for a McAlister subsemigroup",
+[IsMcAlisterTripleSubsemigroup], LambdaBound);
+
 InstallMethod(LambdaBound, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], S ->
 function(r)
@@ -705,6 +856,15 @@ InstallMethod(RhoIdentity, "for a Rees 0-matrix semigroup",
     return ();
   end);
 
+InstallMethod(LambdaIdentity, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+  S -> function(r)
+    return ();
+  end);
+
+InstallMethod(RhoIdentity, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], LambdaIdentity);
+
 InstallMethod(LambdaIdentity, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], S ->
 function(r)
@@ -739,6 +899,12 @@ function(x, y)
   return x![2] ^ -1 * y![2];
 end);
 
+InstallMethod(LambdaPerm, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  return {x, y} -> x[2] ^ -1 * y[2];
+end);
+
 # Returns a permutation mapping LambdaFunc(S)(x) to LambdaFunc(S)(y) so that
 # yx ^ -1(i) = p(i) when RhoFunc(S)(x) = RhoFunc(S)(y)!!
 
@@ -763,7 +929,20 @@ InstallMethod(LambdaConjugator, "for a bipartition semigroup",
 InstallMethod(LambdaConjugator, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], S ->
 function(x, y)
-  return ();
+  return ();  # FIXME is this right???? This is not right!!
+end);
+
+InstallMethod(LambdaConjugator, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  local T, G, act;
+  T := MTSEParent(Representative(S));
+  G := McAlisterTripleSemigroupGroup(T);
+  act := MTSUnderlyingAction(T);  # MTSAction is not an action, causes problems
+  return {x, y} -> RepresentativeAction(G,
+                                        LambdaFunc(S)(x),
+                                        LambdaFunc(S)(y),
+                                        act);
 end);
 
 InstallMethod(LambdaConjugator, "for a matrix semigroup",
@@ -801,6 +980,12 @@ function(j, i)
   return Matrix(parent)[j][i] <> 0;
 end);
 
+InstallMethod(IdempotentTester, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S ->
+function(x, y)
+  return x = y;
+end);
+
 InstallMethod(IdempotentTester, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], S -> function(x, y)
     return MatrixOverFiniteFieldIdempotentTester(S, x, y);
@@ -830,6 +1015,14 @@ function(j, i)
                    [i, mat[j][i] ^ -1, j, mat]);
 end);
 
+InstallMethod(IdempotentCreator, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S ->
+function(x, y)
+  local T;
+  T := MTSEParent(Representative(S));
+  return MTSE(T, x, One(McAlisterTripleSemigroupGroup(T)));
+end);
+
 InstallMethod(IdempotentCreator, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], S -> function(x, y)
     return MatrixOverFiniteFieldIdempotentCreator(S, x, y);
@@ -853,13 +1046,20 @@ InstallMethod(StabilizerAction, "for a bipartition semigroup",
 InstallMethod(StabilizerAction, "for a Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup], S ->
 function(x, p)
-
   if x![1] = 0 then
     return x;
   fi;
   return Objectify(TypeObj(x), [x![1], x![2] * p, x![3], x![4]]);
 end);
 
+InstallMethod(StabilizerAction, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup], S ->
+function(x, p)
+  local T;
+  T := MTSEParent(x);
+  return MTSE(T, x[1], x[2] * p);
+end);
+
 InstallMethod(StabilizerAction, "for a matrix semigroup",
 [IsMatrixOverFiniteFieldSemigroup], S ->
 function(x, y)
@@ -886,6 +1086,10 @@ InstallMethod(IsActingSemigroupWithFixedDegreeMultiplication,
 "for an acting Rees 0-matrix subsemigroup",
 [IsReesZeroMatrixSubsemigroup and IsActingSemigroup], ReturnFalse);
 
+InstallMethod(IsActingSemigroupWithFixedDegreeMultiplication,
+"for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup and IsActingSemigroup], ReturnFalse);
+
 InstallTrueMethod(IsActingSemigroupWithFixedDegreeMultiplication,
                   IsMatrixOverFiniteFieldSemigroup);
 
@@ -913,6 +1117,14 @@ function(S)
   end;
 end);
 
+InstallMethod(SchutzGpMembership, "for a McAlister triple subsemigroup",
+[IsMcAlisterTripleSubsemigroup],
+function(S)
+  return function(stab, x)
+    return SiftedPermutation(stab, x) = ();
+  end;
+end);
+
 InstallMethod(SchutzGpMembership, "for a bipartition semigroup",
 [IsBipartitionSemigroup],
 function(S)
@@ -941,6 +1153,10 @@ InstallMethod(FakeOne, "for a bipartition collection",
 InstallMethod(FakeOne, "for a Rees 0-matrix semigroup element collection",
 [IsReesZeroMatrixSemigroupElementCollection], R -> SEMIGROUPS.UniversalFakeOne);
 
+InstallMethod(FakeOne, "for a McAlister triple semigroup element collection",
+[IsMcAlisterTripleSemigroupElementCollection],
+S -> SEMIGROUPS.UniversalFakeOne);
+
 # Matrix semigroup elements
 InstallMethod(FakeOne, "for an FFE coll coll coll",
 [IsFFECollCollColl], One);

gap/semigroups/semieunit.gd

@@ -28,11 +28,14 @@ DeclareRepresentation("IsMcAlisterTripleSemigroupElementRep",
                       and IsPositionalObjectRep, 3);
 
 DeclareCategoryCollections("IsMcAlisterTripleSemigroupElement");
-DeclareSynonym("IsMTSE", IsMcAlisterTripleSemigroupElement);
+DeclareSynonymAttr("IsMTSE", IsMcAlisterTripleSemigroupElement);
 DeclareSynonymAttr("IsMcAlisterTripleSemigroup",
                    IsInverseSemigroup and IsGeneratorsOfInverseSemigroup
                    and IsMcAlisterTripleSemigroupElementCollection
-                   and IsWholeFamily and IsEnumerableSemigroupRep);
+                   and IsWholeFamily and IsActingSemigroup);
+DeclareSynonymAttr("IsMTS", IsMcAlisterTripleSemigroup);
+DeclareSynonym("IsMcAlisterTripleSubsemigroup",
+IsMcAlisterTripleSemigroupElementCollection and IsSemigroup);
 
 # This is a representation for McAlister triple semigroup, which are
 # created via the function McAlisterTripleSemigroup.
@@ -43,6 +46,9 @@ DeclareSynonymAttr("IsMcAlisterTripleSemigroup",
 #   McAlisterTripleSemigroupPartialOrder
 #   McAlisterTripleSemigroupSemilattice
 #   McAlisterTripleSemigroupAction
+#   McAlisterTripleSemigroupUnderlyingAction
+#   McAlisterTripleSemigroupActionHomomorphism
+#   GeneratorsOfSemigroup
 #
 # their purpose is described in the section of the user manual on McAlister
 # triple semigroups.
@@ -64,19 +70,41 @@ DeclareOperation("McAlisterTripleSemigroup",
 DeclareOperation("McAlisterTripleSemigroup",
                  [IsPermGroup, IsDigraph, IsHomogeneousList]);
 
-# Attributes for McAlister triple semigroups
+# Attributes for McAlister triple subsemigroups
 DeclareAttribute("McAlisterTripleSemigroupGroup",
-                 IsMcAlisterTripleSemigroup and IsWholeFamily);
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSGroup", McAlisterTripleSemigroupGroup);
 DeclareAttribute("McAlisterTripleSemigroupAction",
-                 IsMcAlisterTripleSemigroup and IsWholeFamily);
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSAction", McAlisterTripleSemigroupAction);
 DeclareAttribute("McAlisterTripleSemigroupPartialOrder",
-                 IsMcAlisterTripleSemigroup and IsWholeFamily);
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSPartialOrder", McAlisterTripleSemigroupPartialOrder);
 DeclareAttribute("McAlisterTripleSemigroupSemilattice",
-                 IsMcAlisterTripleSemigroup and IsWholeFamily);
-DeclareAttribute("McAlisterTripleSemigroupElmList",
-                 IsMcAlisterTripleSemigroup and IsWholeFamily);
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSSemilattice", McAlisterTripleSemigroupSemilattice);
+DeclareAttribute("McAlisterTripleSemigroupActionHomomorphism",
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSActionHomomorphism",
+                 McAlisterTripleSemigroupActionHomomorphism);
+DeclareAttribute("McAlisterTripleSemigroupUnderlyingAction",
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSUnderlyingAction",
+                 McAlisterTripleSemigroupUnderlyingAction);
+DeclareAttribute("McAlisterTripleSemigroupSemilatticeVertexLabelInverseMap",
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSSemilatticeVertexLabelInverseMap",
+                 McAlisterTripleSemigroupSemilatticeVertexLabelInverseMap);
 DeclareAttribute("OneImmutable",
                  IsMcAlisterTripleSemigroup and IsWholeFamily and IsMonoid);
+DeclareAttribute("McAlisterTripleSemigroupComponents",
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSComponents",
+                 McAlisterTripleSemigroupComponents);
+DeclareAttribute("McAlisterTripleSemigroupQuotientDigraph",
+                 IsMcAlisterTripleSubsemigroup);
+DeclareSynonymAttr("MTSQuotientDigraph",
+                 McAlisterTripleSemigroupQuotientDigraph);
 
 # Operations for relating to McAlister triple semigroups
 DeclareAttribute("IsomorphismMcAlisterTripleSemigroup",
@@ -91,7 +119,7 @@ DeclareSynonym("MTSE", McAlisterTripleSemigroupElement);
 # Operations for McAlister triple semigroup elements
 DeclareAttribute("McAlisterTripleSemigroupElementParent",
                  IsMcAlisterTripleSemigroupElementRep);
-DeclareSynonym("MTSEParent", McAlisterTripleSemigroupElementParent);
+DeclareSynonymAttr("MTSEParent", McAlisterTripleSemigroupElementParent);
 DeclareOperation("ELM_LIST", [IsMcAlisterTripleSemigroupElementRep, IsPosInt]);
 
 # Inverse semigroup methods

tst/standard/semieunit.tst

@@ -101,26 +101,43 @@ documentation for more detail,
 gap> ps := InverseSemigroup([PartialPerm([2, 3, 4, 5], [1, 3, 5, 4]),
 > PartialPerm([2, 3, 4, 5], [1, 4, 5, 3])]);;
 gap> Mps := IsomorphismSemigroup(IsMcAlisterTripleSemigroup, ps);;
-gap> Image(Mps);
-[ (1, ()), (1, (2,3)), (1, (1,2)), (1, (1,2,3)), (1, (1,3,2)), (1, (1,3)), 
-  (2, ()), (2, (2,3)), (2, (1,2,3)), (2, (1,3)), (3, ()), (3, (2,3)), 
-  (3, (1,2)), (3, (1,3,2)) ]
+gap> Range(Mps);
+<McAlister triple semigroup over Group([ (1,5,6)(2,3,4), (1,4)(2,6)(3,5) ])>
 gap> AsSemigroup(IsMcAlisterTripleSemigroup, ps);
-<McAlister triple semigroup over Group([ (2,3), (1,2,3) ])>
+<McAlister triple semigroup over Group([ (1,5,6)(2,3,4), (1,4)(2,6)(3,5) ])>
 gap> ps := InverseSemigroup([PartialPerm([1, 4, 6, 7], [1, 4, 6, 7]),
 >   PartialPerm([2, 3, 6, 7], [2, 3, 6, 7]), PartialPerm([6, 7], [6, 7]),
 >   PartialPerm([2, 3, 5, 6, 7], [2, 3, 5, 6, 7]),
 >   PartialPerm([1, 4, 6, 7], [2, 3, 7, 6]),
 >   PartialPerm([2, 3, 6, 7], [1, 4, 7, 6]), PartialPerm([6, 7], [7, 6])]);;
 gap> Mps := IsomorphismSemigroup(IsMcAlisterTripleSemigroup, ps);;
-gap> Image(Mps);
-[ (1, ()), (1, (1,2)), (2, ()), (3, ()), (3, (1,2)), (5, ()), (5, (1,2)) ]
-gap> Elements(Range(Mps));
-[ (1, ()), (1, (1,2)), (2, ()), (3, ()), (3, (1,2)), (5, ()), (5, (1,2)) ]
-gap> IsWholeFamily(Image(Mps));
+gap> Range(Mps);
+<McAlister triple semigroup over Group([ (1,2) ])>
+gap> Elements(Range(Mps));;
+gap> IsWholeFamily(Range(Mps));
 true
 gap> AsSemigroup(IsMcAlisterTripleSemigroup, ps);
 <McAlister triple semigroup over Group([ (1,2) ])>
+gap> G := Semigroup(PartialPerm([1, 2, 3], [2, 3, 1]));;
+gap> iso := IsomorphismSemigroup(IsMcAlisterTripleSemigroup, G);;
+gap> PartialPerm([1, 2, 3], [2, 3, 1]) ^ iso;;
+
+#T# McAlister triple subsemigroup methods
+gap> S := Semigroup(Elements(Range(Mps)){[1, 2, 3]});
+<McAlister triple subsemigroup over Group([ (1,2) ])>
+gap> attr := [MTSSemilattice, MTSGroup, MTSPartialOrder, MTSAction,
+> MTSActionHomomorphism, MTSUnderlyingAction, MTSComponents,
+> MTSQuotientDigraph, MTSSemilatticeVertexLabelInverseMap];;
+gap> M := Range(Mps);;
+gap> ForAll(attr, A -> A(S) = A(M));
+true
+gap> Print(Semigroup(Elements(M1){[1, 2, 3]}), "\n");
+Semigroup([ MTSE(McAlisterTripleSemigroup(SymmetricGroup( [ 2 .. 5 ] ), Digrap\
+h( [ [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ), [ 1 .. 4 ]), 1, ()), M\
+TSE(McAlisterTripleSemigroup(SymmetricGroup( [ 2 .. 5 ] ), Digraph( [ [ 1 ], [\
+ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ), [ 1 .. 4 ]), 1, (4,5)), MTSE(McAlis\
+terTripleSemigroup(SymmetricGroup( [ 2 .. 5 ] ), Digraph( [ [ 1 ], [ 1, 2 ], [\
+ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ), [ 1 .. 4 ]), 1, (3,4)) ]
 
 #T# AsSemigroup with bad input
 gap> T := Semigroup([PartialPerm([1], [3]),
@@ -131,26 +148,25 @@ the semigroup is not E-unitary,
 
 #T# Other McAlisterTripleSemigroup tests
 gap> G := SymmetricGroup([2 .. 5]);;
-gap> x := Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 5]]);
-<digraph with 5 vertices, 9 edges>
-gap> y := Digraph([[1], [1, 2], [1, 3], [1, 4]]);
-<digraph with 4 vertices, 7 edges>
+gap> x := Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 5]]);;
+gap> y := Digraph([[1], [1, 2], [1, 3], [1, 4]]);;
 gap> M := McAlisterTripleSemigroup(G, x, y, OnPoints);
 <McAlister triple semigroup over Sym( [ 2 .. 5 ] )>
 gap> IsIsomorphicSemigroup(M, McAlisterTripleSemigroup(G, x, x));
 false
 gap> IsInverseSemigroup(Semigroup(GeneratorsOfSemigroup(M)));
 true
 gap> elms := Enumerator(M);;
-gap> String(elms[1]);
-"MTSE(McAlisterTripleSemigroup(SymmetricGroup( [ 2 .. 5 ] ), Digraph( [ [ 1 ],\
- [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ] ] ), [ 1 .. 4 ]), 1, ())"
+gap> String(elms[1]){[1 .. 40]};
+"MTSE(McAlisterTripleSemigroup(SymmetricG"
 gap> OneImmutable(M);
-Error, Semigroups: OneImutable (for McAlister triple semigroup): usage,
-the argument must be a monoid,
+fail
 gap> M1 := McAlisterTripleSemigroup(G, x, [1, 2]);;
 gap> OneImmutable(M1);
 (2, ())
+gap> x1 := DigraphFromDiSparse6String(".P__@_@_@__D_D_D__H_H_H_@DH_@DHL_@DHL_@DHLp?`abcdefghijklmno");;
+gap> y1 := InducedSubdigraph(x1, [1, 2, 3, 4, 6, 10, 11, 14, 15]);;
+gap> McAlisterTripleSemigroup(AutomorphismGroup(x1), x1, y1);;
 
 #T# McAlister triple semigroup elements
 gap> MTSE(M, 4, (2, 4)(3, 5)) * MTSE(M, 4, (2, 5, 3, 4));
@@ -161,6 +177,10 @@ gap> M = MTSEParent(MTSE(M, 1, (4, 5)));
 true
 gap> M = McAlisterTripleSemigroupElementParent(MTSE(M, 1, (4, 5)));
 true
+gap> LeftOne(MTSE(M, 4, (2, 4)(3, 5))) = MTSE(M, 4, ());
+true
+gap> RightOne(MTSE(M, 4, (2, 4)(3, 5))) = MTSE(M, 2, ());
+true
 gap> MTSE(M, 10, (2, 3, 4, 5));
 Error, Semigroups: McAlisterTripleSemigroupElement: usage,
 second argument should be a vertex label of the join-semilattice of the McAlis\
@@ -214,7 +234,7 @@ gap> M7 := McAlisterTripleSemigroup(Group((5, 6)), x4, x4);;
 gap> IsomorphismSemigroups(M6, M7);
 fail
 
-#T# IsomorphicSemigroups with bad input
+#T# IsomorphismSemigroups with bad input
 gap> x1 := Digraph([[1], [1, 2], [1, 3]]);;
 gap> G := Group((2, 3));;
 gap> M1 := McAlisterTripleSemigroup(G, x1, x1);;
@@ -239,8 +259,7 @@ fail
 #T# IsomorphismSemigroups, where RepresentativeAction fails
 gap> gr := DigraphFromDigraph6String("+H_A?GC_Q@G~wA?G");
 <digraph with 9 vertices, 20 edges>
-gap> G := Group((1, 2, 3)(4, 5, 6), (8, 9));
-Group([ (1,2,3)(4,5,6), (8,9) ])
+gap> G := Group((1, 2, 3)(4, 5, 6), (8, 9));;
 gap> S1 := McAlisterTripleSemigroup(G, gr, [1, 4, 5, 7, 8]);
 <McAlister triple semigroup over Group([ (1,2,3)(4,5,6), (8,9) ])>
 gap> S2 := McAlisterTripleSemigroup(G, gr, [3, 6, 7, 8, 9]);
@@ -289,7 +308,8 @@ gap> S := McAlisterTripleSemigroup(Group((4, 5)),
 gap> IsFInverseSemigroup(S);
 false
 
-#T# EUnitaryInverseCover
+#T# EUnitaryInverseCover 
+#TODO: Add checks that these covers are idempotent separating homomorphisms
 gap> S := InverseMonoid([PartialPermNC([1, 3], [1, 3]),
 > PartialPermNC([1, 2], [3, 1]), PartialPermNC([1, 2], [3, 2])]);;
 gap> cov := EUnitaryInverseCover(S);;
@@ -311,6 +331,13 @@ gap> S := Semigroup([Bipartition([[1, 3, -1, -2, -3], [2]]),
 gap> EUnitaryInverseCover(S);
 Error, Semigroups: EUnitaryInverseCover: usage,
 the argument must be an inverse semigroup,
+gap> S := InverseSemigroup([PartialPerm([1, 2, 4], [4, 3, 2]),
+> PartialPerm([1, 3], [3, 4])]);;
+gap> cov := EUnitaryInverseCover(S);;
+gap> IsEUnitaryInverseSemigroup(Source(cov));
+true
+gap> S = Range(cov);
+true
 
 #T# SEMIGROUPS_UnbindVariables
 gap> Unbind(A);

tst/standard/setup.tst

@@ -29,6 +29,9 @@ gap> IsGeneratorsOfActingSemigroup(Elements(R));
 true
 gap> IsGeneratorsOfActingSemigroup(SLM(2, 2));
 true
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3]), Digraph([[1], [1, 2], [1, 3]]), [1, 2]);;
+gap> IsGeneratorsOfActingSemigroup(M);
+true
 
 # ActionDegree
 
@@ -55,6 +58,11 @@ gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(3), [[()]]);;
 gap> Set(R, ActionDegree);
 [ 0, 1, 3, 4 ]
 
+# ActionDegree, for a MTS element
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3]), Digraph([[1], [1, 2], [1, 3]]), [1, 2]);;
+gap> ActionDegree(M.1);
+0
+
 # ActionDegree, for a matrix over finite field object
 gap> ActionDegree(Matrix(GF(2 ^ 2),
 >                        [[Z(2) ^ 0, 0 * Z(2)], [0 * Z(2), 0 * Z(2)]]));
@@ -85,6 +93,11 @@ gap> ActionDegree([R.1, MultiplicativeZero(R)]);
 gap> ActionDegree([MultiplicativeZero(R)]);
 0
 
+# ActionDegree, for an MTS element collection
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3]), Digraph([[1], [1, 2], [1, 3]]), [1, 2]);;
+gap> ActionDegree(Generators(M));
+0
+
 # ActionDegree, for a matrix object collection
 gap> ActionDegree([Matrix(GF(2),
 >                         [[0 * Z(2), 0 * Z(2)], [0 * Z(2), 0 * Z(2)]]),
@@ -116,6 +129,11 @@ gap> ActionDegree(R);
 gap> ActionDegree(Semigroup(MultiplicativeZero(R)));
 0
 
+# ActionDegree, for an MTS subsemigroup
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3]), Digraph([[1], [1, 2], [1, 3]]), [1, 2]);;
+gap> ActionDegree(Semigroup(Representative(M)));
+0
+
 # ActionDegree, for a matrix over finite field semigroup
 gap> ActionDegree(GLM(2, 2));
 2
@@ -171,6 +189,16 @@ gap> rank(RMSElement(R, 1, (2, 3), 1));
 gap> rank(MultiplicativeZero(R));
 0
 
+# ActionRank, for an MTS semigroup and subsemigroup
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3]), Digraph([[1], [1, 2], [1, 3]]), [1, 2]);;
+gap> rank := ActionRank(M);;
+gap> rank(Representative(M));
+0
+gap> S := Semigroup(Representative(M));;
+gap> rank := ActionRank(S);;
+gap> rank(Representative(S));
+0
+
 # ActionRank, for a matrix over FF
 gap> x := Matrix(GF(2), [[0 * Z(2), 0 * Z(2)], [0 * Z(2), Z(2) ^ 0]]);;
 gap> ActionRank(x, 10);
@@ -203,6 +231,11 @@ gap> R := ReesZeroMatrixSemigroup(Group([()]), [[(), (), 0], [(), (), ()],
 gap> MinActionRank(R);
 0
 
+# MinActionRank, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> MinActionRank(M);
+1
+
 # MinActionRank for a matrix over FF semigroup
 gap> MinActionRank(GLM(2, 2));
 0
@@ -235,6 +268,13 @@ rec(  )
 gap> RhoOrbOpts(R);
 rec(  )
 
+# Rho/lambdaOrbOpts, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> LambdaOrbOpts(M);
+rec(  )
+gap> RhoOrbOpts(M);
+rec(  )
+
 # Rho/LambdaOrbOpts for a matrix over FF semigroup
 gap> LambdaOrbOpts(GLM(2, 2));
 rec(  )
@@ -305,6 +345,37 @@ gap> x(1, s);
 gap> x(2, s);
 1
 
+# Rho/LambdaAct, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> r := MTSE(M, 1, (3, 4));;
+gap> s := MTSE(M, 3, (2, 3));;
+gap> x := LambdaAct(M);;
+gap> x(3, s);
+2
+gap> x(2, s);
+1
+gap> x(2, r);
+1
+gap> x(3, r);
+1
+gap> x(1, r);
+1
+gap> x(0, r);
+1
+gap> x := RhoAct(M);;
+gap> x(3, s);
+1
+gap> x(2, s);
+3
+gap> x(2, r);
+1
+gap> x(3, r);
+1
+gap> x(1, r);
+1
+gap> x(0, r);
+1
+
 # Rho/LambdaAct, for a matrix over FF semigroup
 gap> r := Matrix(GF(2), [[Z(2) ^ 0, Z(2) ^ 0], [Z(2) ^ 0, 0 * Z(2)]]);;
 gap> s := Matrix(GF(2), [[Z(2) ^ 0, Z(2) ^ 0], [0 * Z(2), 0 * Z(2)]]);;
@@ -347,6 +418,13 @@ gap> LambdaOrbSeed(R);
 gap> RhoOrbSeed(R);
 -1
 
+# Rho/LambdaOrbSeed, for an MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> LambdaOrbSeed(M);
+0
+gap> RhoOrbSeed(M);
+0
+
 # Rho/LambdaOrbSeed, for a matrix over FF semigroup
 gap> LambdaOrbSeed(GLM(2, 2));
 <rowbasis of rank 3 over GF(2)>
@@ -399,6 +477,19 @@ gap> x(MultiplicativeZero(S));
 gap> x(RMSElement(S, 1, (1, 3), 2));
 1
 
+# Rho/LambdaFunc, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := LambdaFunc(M);;
+gap> x(MTSE(M, 1, ()));
+1
+gap> x(MTSE(M, 2, (2, 3)));
+3
+gap> x := RhoFunc(M);;
+gap> x(MTSE(M, 1, ()));
+1
+gap> x(MTSE(M, 2, (2, 3)));
+2
+
 # Rho/LambdaFunc, for a matrix over FF semigroup
 gap> S := GLM(2, 3);;
 gap> x := LambdaFunc(S);;
@@ -454,6 +545,23 @@ gap> x(0);
 gap> x(1);
 4
 
+# Rho/LambdaRank, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := LambdaRank(M);;
+gap> x(1);
+2
+gap> x(2);
+1
+gap> x(0);
+0
+gap> x := RhoRank(M);;
+gap> x(1);
+2
+gap> x(2);
+1
+gap> x(0);
+0
+
 # Rho/LambdaRank, for a matrix over FF semigroup
 gap> S := GLM(2, 3);;
 gap> b := NewRowBasisOverFiniteField(IsPlistRowBasisOverFiniteFieldRep,
@@ -517,6 +625,19 @@ gap> x(0, S.1);
 gap> x(2, S.1);
 (2,(),1)
 
+# Rho/LambdaInverse, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := LambdaInverse(M);;
+gap> x(2, MTSE(M, 1, ()));
+(1, ())
+gap> x(2, MTSE(M, 2, (2, 3)));
+(3, (2,3))
+gap> x := RhoInverse(M);;
+gap> x(2, MTSE(M, 1, ()));
+(1, ())
+gap> x(2, MTSE(M, 2, (2, 3)));
+(3, (2,3))
+
 # Rho/LambdaInverse, for a matrix over FF semigroup
 gap> S := GLM(2, 2);;
 gap> x := LambdaInverse(S);;
@@ -573,6 +694,17 @@ infinity
 gap> RhoBound(S)(5);
 120
 
+# Rho/LambdaBound, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> LambdaBound(M)(5);
+6
+gap> LambdaBound(M)(10000);
+6
+gap> RhoBound(M)(5);
+6
+gap> RhoBound(M)(10000);
+6
+
 # Rho/LambdaBound, for a matrix over FF semigroup
 gap> S := GLM(2, 2);;
 gap> LambdaBound(S)(1000);
@@ -618,6 +750,13 @@ gap> LambdaIdentity(S)(2);
 gap> RhoIdentity(S)(2);
 ()
 
+# Rho/LambdaIdentity, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> LambdaIdentity(M)(2);
+()
+gap> RhoIdentity(M)(2);
+()
+
 # Rho/LambdaIdentity, for a matrix over FF semigroup
 gap> S := SLM(2, 2);;
 gap> LambdaIdentity(S)(2);
@@ -651,6 +790,14 @@ gap> x(RMSElement(R, 1, (1, 3, 2), 1), RMSElement(R, 1, (1, 2, 3), 1));
 gap> x(MultiplicativeZero(R), MultiplicativeZero(R));
 ()
 
+# LambdaPerm, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := LambdaPerm(M);;
+gap> x(MTSE(M, 1, (2, 3, 4)), MTSE(M, 2, (2, 3)));
+(2,4)
+gap> x(MTSE(M, 2, ()), MTSE(M, 2, (2, 3)));
+(2,3)
+
 # LambdaPerm, for a matrix over FF semigroup
 gap> x := LambdaPerm(GLM(2, 3));;
 gap> x(Matrix(GF(3), [[Z(3) ^ 0, Z(3) ^ 0], [0 * Z(3), 0 * Z(3)]]),
@@ -681,6 +828,16 @@ gap> x := LambdaConjugator(R);;
 gap> x(RMSElement(R, 1, (1, 3, 2), 1), RMSElement(R, 1, (1, 2, 3), 2));
 ()
 
+# LambdaConjugator, for an MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := LambdaConjugator(M);;
+gap> x(MTSE(M, 1, (2, 3, 4)), MTSE(M, 2, (2, 3)));
+fail
+gap> x(MTSE(M, 2, ()), MTSE(M, 2, (2, 3)));
+(2,3)
+gap> x(MTSE(M, 3, ()), MTSE(M, 2, (2, 3)));
+()
+
 # LambdaConjugator, for a matrix over FF semigroup
 gap> x := LambdaConjugator(GLM(2, 3));;
 gap> x(Matrix(GF(3), [[Z(3) ^ 0, 0 * Z(3)], [0 * Z(3), Z(3) ^ 0]]),
@@ -757,6 +914,31 @@ true
 gap> y(2, 2);
 (2,(1,2),2)
 
+# IdempotentTester and IdempotentCreator, for an MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := IdempotentTester(M);;
+gap> y := IdempotentCreator(M);;
+gap> x(1, 1);
+true
+gap> x(2, 1);
+false
+gap> x(2, 2);
+true
+gap> x(3, 2);
+false
+gap> x(3, 3);
+true
+gap> y(2, 2);
+(2, ())
+gap> y(1, 2);
+(1, ())
+gap> y(1, 1);
+(1, ())
+gap> y(3, 2);
+(3, ())
+gap> y(3, 3);
+(3, ())
+
 # IdempotentTester and IdempotentCreator, for a matrix over FF semigroup
 gap> S := GLM(2, 3);;
 gap> x := IdempotentTester(S);;
@@ -804,6 +986,16 @@ gap> x(MultiplicativeZero(R), ());
 gap> x(RMSElement(R, 1, (), 1), ());
 (1,(),1)
 
+# StabilizerAction, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> x := StabilizerAction(M);;
+gap> x(MTSE(M, 1, ()), ());
+(1, ())
+gap> x(MTSE(M, 2, (2, 3)), (2, 3));
+(2, ())
+gap> x(MTSE(M, 3, ()), (2, 4, 3));
+(3, (2,4,3))
+
 # StabilizerAction, for a matrix over FF semigroup
 gap> S := GLM(2, 3);;
 gap> x := StabilizerAction(S);;
@@ -832,6 +1024,11 @@ gap> S := ReesZeroMatrixSemigroup(Group(()), [[()]]);;
 gap> IsActingSemigroupWithFixedDegreeMultiplication(Semigroup(S));
 false
 
+# IsActingSemigroupWithFixedDegreeMultiplication, for a MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> IsActingSemigroupWithFixedDegreeMultiplication(Semigroup(M));
+false
+
 # IsActingSemigroupWithFixedDegreeMultiplication, for a matrix over FF semigroup
 gap> IsActingSemigroupWithFixedDegreeMultiplication(
 > GLM(2, 2));
@@ -870,11 +1067,21 @@ true
 # SchutzGpMembership, for an RZMS
 gap> R := ReesZeroMatrixSemigroup(Group((1, 2, 3)), [[()]]);;
 gap> R := Semigroup(Elements(R));;
-gap> o := LambdaOrb(S);; Enumerate(o);;
-gap> schutz := LambdaOrbStabChain(o, 2);;
+gap> o := LambdaOrb(R);; Enumerate(o);;
+gap> schutz := LambdaOrbStabChain(o, 3);;
 gap> SchutzGpMembership(R)(schutz, ());
 true
 
+# SchutzGpMembership, for an MTS
+gap> M := McAlisterTripleSemigroup(SymmetricGroup([2, 3, 4]), Digraph([[1], [1, 2], [1, 3], [1, 4]]), [1, 2, 3]);;
+gap> M := Semigroup(Elements(M));;
+gap> o := LambdaOrb(M);; Enumerate(o);;
+gap> schutz := LambdaOrbStabChain(o, 3);;
+gap> SchutzGpMembership(M)(schutz, ());
+true
+
+# gap> SchutzGpMembership(M)(schutz, ()); TODO: THIS DOESN'T WORK!
+
 # SchutzGpMembership, for a matrix over FF semigroup
 gap> S := Monoid([
 >  Matrix(GF(2), [[0 * Z(2), Z(2) ^ 0], [0 * Z(2), 0 * Z(2)]]),
@@ -903,6 +1110,10 @@ gap> FakeOne(PartitionMonoid(1));
 gap> FakeOne(ReesZeroMatrixSemigroup(Group(()), [[()]]));
 <universal fake one>
 
+# FakeOne, for a MTS
+gap> FakeOne(McAlisterTripleSemigroup(Group(()), Digraph([[1]]), [1]));
+<universal fake one>
+
 # FakeOne, for a matrix over FF semigroup
 gap> FakeOne(GLM(2, 2));
 Matrix(GF(2), [[Z(2)^0, 0*Z(2)], [0*Z(2), Z(2)^0]])