diff --git a/doc/semieunit.xml b/doc/semieunit.xml
index a74e4e39be..07f499c9f0 100644
--- a/doc/semieunit.xml
+++ b/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>
diff --git a/gap/main/setup.gi b/gap/main/setup.gi
index d354d97bf5..772ef310b5 100644
--- a/gap/main/setup.gi
+++ b/gap/main/setup.gi
@@ -51,7 +51,8 @@ InstallMethod(IsGeneratorsOfActingSemigroup,
 "for a McAlister triple element collection",
 [IsMcAlisterTripleSemigroupElementCollection],
 function(coll)
-  return IsPermGroup(McAlisterTripleSemigroupGroup(MTSEParent(coll[1])));
+  return
+  IsPermGroup(McAlisterTripleSemigroupGroup(MTSEParent(Representative(coll))));
     # and McAlisterTripleSemigroupAction(MTSEParent(coll[1])) = OnPoints;
 end);
 
@@ -102,11 +103,10 @@ function(x)
   return NrMovedPoints(x![2]) + 1;
 end);
 
-# TODO correct?
 InstallMethod(ActionDegree, "for a McAlister semigroup element",
 [IsMcAlisterTripleSemigroupElement],
 function(x)
-  return NrMovedPoints(x[2]);
+  return 0;
 end);
 
 InstallMethod(ActionDegree, "for a matrix over finite field object",
@@ -169,14 +169,10 @@ function(R)
   return 0;
 end);
 
-# TODO check this is ok?
-
 InstallMethod(ActionDegree, "for a McAlister triple subsemigroup",
 [IsMcAlisterTripleSubsemigroup],
 function(S)
-  local rep;
-  rep := Representative(S);
-  return NrMovedPoints(McAlisterTripleSemigroupGroup(MTSEParent(rep)));
+  return 0;
 end);
 
 InstallMethod(ActionDegree, "for a matrix over finite field semigroup",
@@ -248,8 +244,6 @@ function(R)
   end;
 end);
 
-# TODO: ok?
-
 InstallMethod(ActionRank, "for a McAlister triple semigroup element and int",
 [IsMcAlisterTripleSemigroupElement, IsInt],
 function(f, n)
@@ -260,8 +254,6 @@ function(f, n)
   return Position(DigraphTopologicalSort(digraph), id[f[1]]);
 end);
 
-# TODO: ok?
-
 InstallMethod(ActionRank, "for a McAlister triple subsemigroup",
 [IsMcAlisterTripleSubsemigroup],
 function(S)
@@ -784,11 +776,11 @@ end);
 InstallMethod(RhoBound, "for a Rees 0-matrix semigroup",
 [IsReesZeroMatrixSubsemigroup], LambdaBound);
 
-# TODO improve the upper bound here
 InstallMethod(LambdaBound, "for a McAlister subsemigroup",
 [IsMcAlisterTripleSubsemigroup], S ->
 function(r)
-  return Size(McAlisterTripleSemigroupGroup(MTSEParent(Representative(S))));
+  G := McAlisterTripleSemigroupGroup(MTSEParent(Representative(S)));
+  return Size(Stabilizer(G, r[1]));
 end);
 
 InstallMethod(RhoBound, "for a McAlister subsemigroup",
diff --git a/tst/standard/semieunit.tst b/tst/standard/semieunit.tst
index 923ee7f1f4..b7ef35f877 100644
--- a/tst/standard/semieunit.tst
+++ b/tst/standard/semieunit.tst
@@ -101,11 +101,8 @@ 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, (1,2)(3,6)(4,5)), (1, (1,4)(2,6)(3,5)), (1, (1,6,5)(2,4,3)), 
-  (3, ()), (3, (1,3)(2,5)(4,6)), (3, (1,4)(2,6)(3,5)), (3, (1,5,6)(2,3,4)), 
-  (4, ()), (4, (1,2)(3,6)(4,5)), (4, (1,3)(2,5)(4,6)), (4, (1,4)(2,6)(3,5)), 
-  (4, (1,5,6)(2,3,4)), (4, (1,6,5)(2,4,3)) ]
+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([ (1,5,6)(2,3,4), (1,4)(2,6)(3,5) ])>
 gap> ps := InverseSemigroup([PartialPerm([1, 4, 6, 7], [1, 4, 6, 7]),
@@ -114,17 +111,22 @@ gap> ps := InverseSemigroup([PartialPerm([1, 4, 6, 7], [1, 4, 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, ()), (2, (1,2)), (3, ()), (3, (1,2)), (4, ()) ]
-gap> Elements(Range(Mps));
-[ (1, ()), (1, (1,2)), (2, ()), (2, (1,2)), (3, ()), (3, (1,2)), (4, ()) ]
+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);
+MappingByFunction( <partial perm group of size 3, rank 3 with 1 generator>
+, <McAlister triple semigroup over Group([ (2,3,
+4) ])>, function( s ) ... end )
+gap> PartialPerm([1, 2, 3], [2, 3, 1]) ^ iso;;
 
 #T# McAlister triple subsemigroup methods
-gap> S := Semigroup(Image(Mps){[1 .. 3]});
+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,
@@ -132,13 +134,13 @@ gap> attr := [MTSSemilattice, MTSGroup, MTSPartialOrder, MTSAction,
 gap> M := Range(Mps);;
 gap> ForAll(attr, A -> A(S) = A(M));
 true
-gap> Print(S, "\n");
-Semigroup([ MTSE(McAlisterTripleSemigroup(Group( [ (1,2) ] ), Digraph( [ [ 1, \
-3 ], [ 2, 3 ], [ 3 ], [ 2, 3, 4 ], [ 1, 3, 5 ] ] ), [ 3, 2, 4, 1 ]), 1, ()), M\
-TSE(McAlisterTripleSemigroup(Group( [ (1,2) ] ), Digraph( [ [ 1, 3 ], [ 2, 3 ]\
-, [ 3 ], [ 2, 3, 4 ], [ 1, 3, 5 ] ] ), [ 3, 2, 4, 1 ]), 1, (1,2)), MTSE(McAlis\
-terTripleSemigroup(Group( [ (1,2) ] ), Digraph( [ [ 1, 3 ], [ 2, 3 ], [ 3 ], [\
- 2, 3, 4 ], [ 1, 3, 5 ] ] ), [ 3, 2, 4, 1 ]), 2, ()) ]
+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]),
@@ -149,10 +151,8 @@ 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));
@@ -160,14 +160,16 @@ 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 ]), 2, (3,4,5))"
+gap> String(elms[1]){[1 .. 40]};
+"MTSE(McAlisterTripleSemigroup(SymmetricG"
 gap> OneImmutable(M);
 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));
@@ -235,7 +237,7 @@ 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);;
@@ -260,8 +262,7 @@ 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]);
diff --git a/tst/standard/setup.tst b/tst/standard/setup.tst
index 7c376d26f6..9a0ec1451b 100644
--- a/tst/standard/setup.tst
+++ b/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]])