Merge pull request #22 from erich-9/fix-examples

Fix files in directory "examples".

examples/examples_chain_complexes.g

@@ -45,8 +45,8 @@ IsExactInDegree(C,2);
 C;
 Shift(C,1);
 D := Shift(C,-1);
-dc := DifferentialOfComplex(C,3)!.maps;
-dd := DifferentialOfComplex(D,4)!.maps;
+dc := DifferentialOfComplex(C,3);
+dd := DifferentialOfComplex(D,4);
 MatricesOfPathAlgebraMatModuleHomomorphism(dc);
 MatricesOfPathAlgebraMatModuleHomomorphism(dd);
 ####################

examples/examples_homomorphisms.g

@@ -34,9 +34,9 @@ y := ZeroMapping(L,N);
 y = z;            
 id := IdentityMapping(N);  
 f*id;
-#This causes an error!
-id*f;
-quit;;
+# #This causes an error!
+# id*f;
+# quit;;
 2*f + z;
 ###########
 L := RightModuleOverPathAlgebra(A,[["a",[0,1]],["b",[0,1]],
@@ -53,7 +53,7 @@ IsIsomorphism(h);
 S := SimpleModules(A)[1];;
 H := HomOverAlgebra(N,S);; 
 IsSplitMonomorphism(H[1]);  
-f := IsSplitEpimorphism(H[1]);
+IsSplitEpimorphism(H[1]);
 IsSplitMonomorphism(f);
 ###########
 L := RightModuleOverPathAlgebra(A,[["a",[0,1]],["b",[0,1]],

examples/examples_path_algebras.g

@@ -1 +1,130 @@
-Q := Quiver( ["u","v"] , [ ["u","u","a"], ["u","v","b"], ["v","u","c"], ["v","v","d"] ] );F := Rationals;FQ := PathAlgebra(F,Q);##########IsPathAlgebra(FQ);##########QuiverOfPathAlgebra(FQ);##########FQ.a;FQ.v;elem := 2*FQ.a - 3*FQ.v;##########IsLeftUniform(elem);IsRightUniform(elem);IsUniform(elem);another := FQ.a*FQ.b + FQ.b*FQ.d*FQ.c*FQ.b*FQ.d;IsLeftUniform(another);IsRightUniform(another);IsUniform(another);##########elem := FQ.a*FQ.b*FQ.c + FQ.b*FQ.d*FQ.c+FQ.d*FQ.d;LeadingTerm(elem);LeadingCoefficient(elem);mon := LeadingMonomial(elem);mon in FQ;mon in Q;##########Q := Quiver( 1, [ [1,1,"a"], [1,1,"b"] ] );kQ := PathAlgebra(Rationals, Q);gens := GeneratorsOfAlgebra(kQ);a := gens[2];b := gens[3];relations := [a^2,a*b-b*a, b*b];I := Ideal(kQ,relations);A := kQ/I;IndecProjectiveModules(A);  ##########gb := GBNPGroebnerBasis(relations,kQ);I := Ideal(kQ,gb);GroebnerBasis(I,gb);IndecProjectiveModules(A);A := kQ/I;IndecProjectiveModules(A);##########I := Ideal(FQ, [FQ.a - FQ.b*FQ.c, FQ.d*FQ.d]); GeneratorsOfIdeal(I);IsIdealInPathAlgebra(I);##########rels := [FQ.a - FQ.b*FQ.c, FQ.d*FQ.d];gb := GBNPGroebnerBasis(rels, FQ); I := Ideal(FQ, gb);GroebnerBasis(I, gb);quot := FQ/I;##########quot := FQ/I;IsQuotientOfPathAlgebra(quot);IsQuotientOfPathAlgebra(FQ);##########Q := Quiver(5, [ [1,2,"a"], [2,4,"b"], [3,2,"c"], [2,5,"d"] ]);A := PathAlgebra(Rationals, Q);IsFiniteTypeAlgebra(A);quo := A/[A.a*A.b, A.c*A.d];;IsFiniteTypeAlgebra(quo);##########elem := quot.a*quot.b;IsElementOfQuotientOfPathAlgebra(elem);IsElementOfQuotientOfPathAlgebra(FQ.a*FQ.b); ##########alg := NakayamaAlgebra([2,1], Rationals);QuiverOfPathAlgebra(alg);##########Q := Quiver(1, [ [1,1,"a"], [1,1,"b"] ]);;  A := PathAlgebra(Rationals, Q);;IsSpecialBiserialAlgebra(A); IsStringAlgebra(A);rel1 := [A.a*A.b, A.a^2, A.b^2];  quo1 := A/rel1;;IsSpecialBiserialAlgebra(quo1); IsStringAlgebra(quo1);rel2 := [A.a*A.b-A.b*A.a, A.a^2, A.b^2];  quo2 := A/rel2;;IsSpecialBiserialAlgebra(quo2); IsStringAlgebra(quo2);rel3 := [A.a*A.b+A.b*A.a, A.a^2, A.b^2, A.b*A.a];   quo3 := A/rel3;;IsSpecialBiserialAlgebra(quo3); IsStringAlgebra(quo3);rel4 := [A.a*A.b, A.a^2, A.b^3]; quo4 := A/rel4;;IsSpecialBiserialAlgebra(quo4); IsStringAlgebra(quo4);##########Q := Quiver( [ "u", "v" ], [ [ "u", "u", "a" ],              [ "u", "v", "b" ] ] );Qop := OppositeQuiver(Q);VerticesOfQuiver( Qop );ArrowsOfQuiver( Qop );OppositePath( Q.a * Q.b );IsIdenticalObj( Q, OppositeQuiver( Qop ) );OppositePath( Qop.b_op * Qop.a_op );##########Q := Quiver( [ "u", "v" ], [ [ "u", "u", "a" ],              [ "u", "v", "b" ] ] );A := PathAlgebra( Rationals, Q );OppositePathAlgebra( A );OppositePathAlgebraElement( A.u + 2*A.a + 5*A.a*A.b );IsIdenticalObj( A,         OppositePathAlgebra( OppositePathAlgebra( A ) ) );##########q1 := Quiver( [ "u1", "u2" ], [ [ "u1", "u2", "a" ] ] );q2 := Quiver( [ "v1", "v2", "v3" ],                      [ [ "v1", "v2", "b" ],                        [ "v2", "v3", "c" ] ] );q1_q2 := QuiverProduct( q1, q2 );q1_q2.u1_b * q1_q2.a_v2;IncludeInProductQuiver( [ q1.a, q2.b * q2.c ], q1_q2 );ProjectFromProductQuiver( 2, q1_q2.a_v1 * q1_q2.u2_b * q1_q2.u2_c );q1_q2_dec := QuiverProductDecomposition( q1_q2 );q1_q2_dec[ 1 ];q1_q2_dec[ 1 ] = q1;##########q1 := Quiver( [ "u1", "u2" ], [ [ "u1", "u2", "a" ] ] );q2 := Quiver( [ "v1", "v2", "v3", "v4" ],                      [ [ "v1", "v2", "b" ],                        [ "v1", "v3", "c" ],                        [ "v2", "v4", "d" ],                        [ "v3", "v4", "e" ] ] );fq1 := PathAlgebra( Rationals, q1 );fq2 := PathAlgebra( Rationals, q2 );I := Ideal( fq2, [ fq2.b * fq2.d - fq2.c * fq2.e ] );quot := fq2 / I;t := TensorProductOfAlgebras( fq1, quot );SimpleTensor( [ fq1.a, quot.b ], t );t_dec := TensorProductDecomposition( t );t_dec[ 1 ] = fq1;
+Q := Quiver( ["u","v"] , [ ["u","u","a"], ["u","v","b"], 
+["v","u","c"], ["v","v","d"] ] );
+F := Rationals;
+FQ := PathAlgebra(F,Q);
+##########
+IsPathAlgebra(FQ);
+##########
+QuiverOfPathAlgebra(FQ);
+##########
+FQ.a;
+FQ.v;
+elem := 2*FQ.a - 3*FQ.v;
+##########
+IsLeftUniform(elem);
+IsRightUniform(elem);
+IsUniform(elem);
+another := FQ.a*FQ.b + FQ.b*FQ.d*FQ.c*FQ.b*FQ.d;
+IsLeftUniform(another);
+IsRightUniform(another);
+IsUniform(another);
+##########
+elem := FQ.a*FQ.b*FQ.c + FQ.b*FQ.d*FQ.c+FQ.d*FQ.d;
+LeadingTerm(elem);
+LeadingCoefficient(elem);
+mon := LeadingMonomial(elem);
+mon in FQ;
+mon in Q;
+##########
+Q := Quiver( 1, [ [1,1,"a"], [1,1,"b"] ] );
+kQ := PathAlgebra(Rationals, Q);
+gens := GeneratorsOfAlgebra(kQ);
+a := gens[2];
+b := gens[3];
+relations := [a^2,a*b-b*a, b*b];
+A := kQ/relations;
+IndecProjectiveModules(A);  
+##########
+gb := GBNPGroebnerBasis(relations,kQ);
+I := Ideal(kQ,gb);
+GroebnerBasis(I,gb);
+IndecProjectiveModules(A);
+A := kQ/I;
+IndecProjectiveModules(A);
+##########
+I := Ideal(FQ, [FQ.a - FQ.b*FQ.c, FQ.d*FQ.d]); 
+GeneratorsOfIdeal(I);
+IsIdealInPathAlgebra(I);
+##########
+rels := [FQ.a - FQ.b*FQ.c, FQ.d*FQ.d];
+gb := GBNPGroebnerBasis(rels, FQ); 
+I := Ideal(FQ, gb);
+GroebnerBasis(I, gb);
+quot := FQ/I;
+##########
+quot := FQ/I;
+IsQuotientOfPathAlgebra(quot);
+IsQuotientOfPathAlgebra(FQ);
+##########
+Q := Quiver(5, [ [1,2,"a"], [2,4,"b"], [3,2,"c"], [2,5,"d"] ]);
+A := PathAlgebra(Rationals, Q);
+IsFiniteTypeAlgebra(A);
+quo := A/[A.a*A.b, A.c*A.d];;
+IsFiniteTypeAlgebra(quo);
+##########
+elem := quot.a*quot.b;
+IsElementOfQuotientOfPathAlgebra(elem);
+IsElementOfQuotientOfPathAlgebra(FQ.a*FQ.b); 
+##########
+alg := NakayamaAlgebra([2,1], Rationals);
+QuiverOfPathAlgebra(alg);
+##########
+Q := Quiver(1, [ [1,1,"a"], [1,1,"b"] ]);;  
+A := PathAlgebra(Rationals, Q);;
+IsSpecialBiserialAlgebra(A); IsStringAlgebra(A);
+rel1 := [A.a*A.b, A.a^2, A.b^2];  
+quo1 := A/rel1;;
+IsSpecialBiserialAlgebra(quo1); IsStringAlgebra(quo1);
+rel2 := [A.a*A.b-A.b*A.a, A.a^2, A.b^2];  
+quo2 := A/rel2;;
+IsSpecialBiserialAlgebra(quo2); IsStringAlgebra(quo2);
+rel3 := [A.a*A.b+A.b*A.a, A.a^2, A.b^2, A.b*A.a];   
+quo3 := A/rel3;;
+IsSpecialBiserialAlgebra(quo3); IsStringAlgebra(quo3);
+rel4 := [A.a*A.b, A.a^2, A.b^3]; 
+quo4 := A/rel4;;
+IsSpecialBiserialAlgebra(quo4); IsStringAlgebra(quo4);
+##########
+Q := Quiver( [ "u", "v" ], [ [ "u", "u", "a" ], 
+             [ "u", "v", "b" ] ] );
+Qop := OppositeQuiver(Q);
+VerticesOfQuiver( Qop );
+ArrowsOfQuiver( Qop );
+OppositePath( Q.a * Q.b );
+IsIdenticalObj( Q, OppositeQuiver( Qop ) );
+OppositePath( Qop.b_op * Qop.a_op );
+##########
+Q := Quiver( [ "u", "v" ], [ [ "u", "u", "a" ], 
+             [ "u", "v", "b" ] ] );
+A := PathAlgebra( Rationals, Q );
+OppositePathAlgebra( A );
+OppositePathAlgebraElement( A.u + 2*A.a + 5*A.a*A.b );
+IsIdenticalObj( A, 
+        OppositePathAlgebra( OppositePathAlgebra( A ) ) );
+##########
+q1 := Quiver( [ "u1", "u2" ], [ [ "u1", "u2", "a" ] ] );
+q2 := Quiver( [ "v1", "v2", "v3" ],
+                      [ [ "v1", "v2", "b" ],
+                        [ "v2", "v3", "c" ] ] );
+q1_q2 := QuiverProduct( q1, q2 );
+q1_q2.u1_b * q1_q2.a_v2;
+IncludeInProductQuiver( [ q1.a, q2.b * q2.c ], q1_q2 );
+ProjectFromProductQuiver( 2, q1_q2.a_v1 * q1_q2.u2_b * q1_q2.u2_c );
+q1_q2_dec := QuiverProductDecomposition( q1_q2 );
+q1_q2_dec[ 1 ];
+q1_q2_dec[ 1 ] = q1;
+##########
+q1 := Quiver( [ "u1", "u2" ], [ [ "u1", "u2", "a" ] ] );
+q2 := Quiver( [ "v1", "v2", "v3", "v4" ],
+                      [ [ "v1", "v2", "b" ],
+                        [ "v1", "v3", "c" ],
+                        [ "v2", "v4", "d" ],
+                        [ "v3", "v4", "e" ] ] );
+fq1 := PathAlgebra( Rationals, q1 );
+fq2 := PathAlgebra( Rationals, q2 );
+rels := [ fq2.b * fq2.d - fq2.c * fq2.e ];
+quot := fq2 / rels;
+t := TensorProductOfAlgebras( fq1, quot );
+SimpleTensor( [ fq1.a, quot.b ], t );
+t_dec := TensorProductDecomposition( t );
+t_dec[ 1 ] = fq1;

examples/examples_quivers.g

@@ -54,13 +54,13 @@ ArrowsOfQuiver(q1_op);
 Q := Quiver(6, [ [1,2],[1,1],[3,2],[4,5],[4,5] ]);
 VerticesOfQuiver(Q);
 FullSubquiver(Q, [Q.v1, Q.v2]);
-ConnectedComponents(Q);
+ConnectedComponentsOfQuiver(Q);
 ###############
 q1 := Quiver(["u","v"],[["u","u","a"],["u","v","b"],
               ["v","u","c"],["v","v","d"]]);
 IsPath(q1.b);
 IsPath(q1.u);
-IsVertex(q1.c);
+IsQuiverVertex(q1.c);
 IsZeroPath(q1.d);
 ###############
 q1 := Quiver(["u","v"],[["u","u","a"],["u","v","b"],

examples/examples_right_modules.g

@@ -56,7 +56,7 @@ mat := [["a",[[1,2],[0,3],[1,5]]],["b",[[2,0],[3,0],[5,0]]],
 ["c",[[0,0],[1,0]]],["d",[[1,2],[0,1]]],["e",[[0,0,0],[0,0,0]]]];;
 N := RightModuleOverPathAlgebra(A,mat);          
 ##################
-N2 := DirectSumOfModules([N,N]);
+N2 := DirectSumOfQPAModules([N,N]);
 proj := DirectSumProjections(N2);
 inc := DirectSumInclusions(N2);  
 ##################