diff --git a/doc/algebra.xml b/doc/algebra.xml index 33bf8dc..dbacf73 100644 --- a/doc/algebra.xml +++ b/doc/algebra.xml @@ -56,8 +56,7 @@ on an ideal I of A. - - A1 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );; gap> SetName( A1, "A1" ); gap> BA1 := BasisVectors( Basis( A1 ) );; @@ -72,8 +71,7 @@ gap> m1 := RegularAlgebraMultiplier( A1, I1, v1 ); (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> [ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) ] -]]> - +]]> a,b in the basis for A. - - IsAlgebraMultiplier( m1 ); true gap> id1 := One( A1 );; @@ -97,8 +94,7 @@ gap> h1 := LeftModuleHomomorphismByImages( A1, A1, BA1, L1 ); (Z(5)^0)*() ] gap> IsAlgebraMultiplier( h1 ); false -]]> - +]]> \mu_b \circ \mu_{b'} = \mu_{bb'}. - - u1 := BA1[3]; (Z(5)^0)*(1,3,5)(2,4,6) gap> S1 := Subalgebra( A3, [ u1 ] );; @@ -123,8 +118,7 @@ gap> SetName( MS1, "MS1" ); gap> BMS1 := BasisVectors( Basis( MS1 ) );; gap> BMS1[1]; I1> -]]> - +]]> MultiplierAlgebraOfIdealBySubalgebra(A,A,A);. - - MA1 := MultiplierAlgebra( A1 ); gap> BMA1 := BasisVectors( Basis( MA1 ) );; gap> BMA1[3]; -> > -]]> - +]]> B to M mapping b to \mu_b. - - hom1 := MultiplierHomomorphism( MA1 );; gap> ImageElm( hom1, BA1[2] ); Basis( A1, [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2\ @@ -169,8 +160,7 @@ Basis( A1, [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2\ ] ) -> [ (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2), (Z(5)^0)*() ] -]]> - +]]> @@ -242,8 +232,7 @@ as shown in - - A1 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );; gap> BA1 := BasisVectors( Basis( A1 ) );; gap> v := BA1[1] + BA1[3] + BA1[5]; @@ -253,8 +242,7 @@ gap> act1 := AlgebraActionByMultipliers( A1, I1, A1 );; gap> act12 := Image( act1, BA1[2] );; gap> Image( act12, v ); (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) -]]> - +]]> Q_2 has basis \{[m_2],[m_2^2]\}. - - theta1 := NaturalHomomorphismByIdeal( A1, I1 ); -> > @@ -321,8 +308,7 @@ gap> [ Image(b1,m2)=m2^2, Image(b1,m2^2)=m2^3, Image(b1,m2^3)=Zero(A2) ]; [ true, true, true ] gap> [ Image(b2,m2)=m2^3, b2=b1^2 ]; [true, true ] -]]> - +]]> <#Include Label="AlgebraActionByHomomorphism"> @@ -393,8 +379,7 @@ Continuing the example above, - - P1 := SemidirectProductOfAlgebras( A1, act1, I1 ); gap> Embedding( P1, 1 ); @@ -419,8 +404,7 @@ gap> Embedding( P2, 2 ); [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ] -> [ v.3, v.4, v.5 ] -]]> - +]]> - - A2c6 := GroupRing( GF(2), Group( (1,2,3,4,5,6) ) );; gap> R2c3 := GroupRing( GF(2), Group( (7,8,9) ) );; gap> homAR := AllAlgebraHomomorphisms( A2c6, R2c3 );; @@ -496,8 +479,7 @@ gap> List( bijAA, h -> MappingGeneratorsImages(h) ); gap> ideAA := AllIdempotentAlgebraHomomorphisms( A2c6, A2c6 );; gap> Length( ideAA ); 14 -]]> - +]]> diff --git a/doc/cat1.xml b/doc/cat1.xml index dbb0e38..9505568 100644 --- a/doc/cat1.xml +++ b/doc/cat1.xml @@ -121,8 +121,7 @@ constructed in section . - - t4 := homAR[8]; [ (Z(2)^0)*(1,6,5,4,3,2) ] -> [ (Z(2)^0)*(7,9,8) ] gap> e4 := homRA[8]; @@ -157,8 +156,7 @@ Cat1-algebra [..=>..] :- : kernel embedding maps generators of kernel to: [ (Z(2)^0)*()+(Z(2)^0)*(1,4)(2,5)(3,6), (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)* (1,5,3)(2,6,4), (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,6,5,4,3,2) ] -]]> - +]]> - - C := Cat1AlgebraSelect( 11 ); |--------------------------------------------------------| | 11 is invalid value for the Galois Field (GFnum) | @@ -236,8 +233,7 @@ There are 4 cat1-structures for the group algebra GF(2)_c6. Usage: Cat1Algebra( GFnum, gpsize, gpnum, num ); Algebra has generators [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3)(4,5) ] 4 -]]> - +]]> The algebra GF(n)_gp has a list of n^{|gp|} elements. The [2, 10] in the second structure above @@ -245,8 +241,7 @@ indicates that the tail map, and also the head map, of the cat^1-algebra maps the two generators of c6 to the second and tenth elements of this algebra respectively. - - C0 := Cat1AlgebraSelect( 4, 6, 2, 2 ); [GF(2^2)_c6 -> Algebra( GF(2^2), [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)(3,6)+( @@ -279,8 +274,7 @@ Cat1-algebra [GF(2^2)_c6=>..] :- : kernel embedding maps generators of kernel to: [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4) (2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] -]]> - +]]> ^{1}-algebras of a given cat^{1}-algebra. - - C6 := Cat1AlgebraSelect( 2, 6, 2, 4 );; gap> A6 := Source( C6 ); GF(2)_c6 @@ -360,8 +353,7 @@ Cat1-algebra [..=>..] :- [ of ..., (Z(2)^0)*()+(Z(2)^0)*(4,5) ] gap> IsSubCat1Algebra( C6, SC6 ); true -]]> - +]]> @@ -429,8 +421,7 @@ These are the six main attributes of a cat^{1}-algebra morphism. - - C1 := Cat1AlgebraSelect( 2, 1, 1, 1 ); [GF(2)_triv -> GF(2)_triv] gap> Display( C1 ); @@ -513,8 +504,7 @@ gap> IsInjective( m12 ); true gap> IsBijective( m12 ); false -]]> - +]]> - - im12 := ImagesSource2DimensionalMapping( m12 );; gap> Display( im12 ); Cat1-algebra [..=>..] :- @@ -542,8 +531,7 @@ Cat1-algebra [..=>..] :- : range embedding maps range generators to: [ (Z(2)^0)*() ] : the kernel is trivial. -]]> - +]]> diff --git a/doc/convert.xml b/doc/convert.xml index 2a42410..418f936 100644 --- a/doc/convert.xml +++ b/doc/convert.xml @@ -69,8 +69,7 @@ As an example we use the crossed module XAB constructed in section - - Cn := Cat1AlgebraOfXModAlgebra( Xn ); [An |X Bn -> An] gap> Display( Cn ); @@ -107,8 +106,7 @@ Cat1-algebra [An |X Bn => An] :- [ 0*Z(5), 0*Z(5), 0*Z(5) ] ] ] : range embedding maps range generators to: [ v.1, v.2 ] : kernel has generators: [ v.4, v.5 ] -]]> - +]]> <#Include Label="Cat1AlgebraOfXModAlgebra"> @@ -125,8 +123,7 @@ constructed in section . - - X6 := XModAlgebraOfCat1Algebra( C6 ); [ -> ] gap> Display( X6 ); @@ -138,8 +135,7 @@ Crossed module [..->..] :- [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3) ] : Boundary homomorphism maps source generators to: [ of ..., of ..., of ... ] -]]> - +]]> diff --git a/doc/intro.xml b/doc/intro.xml index d553e22..4eab2d9 100644 --- a/doc/intro.xml +++ b/doc/intro.xml @@ -55,11 +55,9 @@ There are aspects of commutative algebras for which no &GAP; functions yet exist We have included here functions for all homomorphisms of algebras.

The package is loaded with the command - - LoadPackage( "xmodalg" ); -]]> - +]]>

The package may be obtained as a compressed .tar file @@ -75,24 +73,20 @@ can be found in the documentation folder. The html versions, with or without MathJax, may be rebuilt as follows:

- - ReadPackage( "xmodalg", "makedoc.g" ); -]]> - +]]>

It is possible to check that the package has been installed correctly by running the test files (this terminates the &GAP; session):

- - TestPackage( "xmodalg" ); Architecture: . . . . . testing: . . . . . . . . #I No errors detected while testing -]]> - +]]> diff --git a/doc/xmod.xml b/doc/xmod.xml index 75d8fd9..04ac385 100644 --- a/doc/xmod.xml +++ b/doc/xmod.xml @@ -76,8 +76,7 @@ it is the case that {\partial(S)} is an ideal of R. - - F5 := GF(5);; gap> id5 := One( F5 );; gap> two := Z(5);; @@ -95,8 +94,7 @@ gap> actn := AlgebraActionByMultipliers( An, Bn, An );; gap> Xn := XModAlgebraByIdeal( An, Bn ); [ Bn -> An ] gap> SetName( Xn, "Xn" ); -]]> - +]]> XModAlgebraByIdeal formed using the - - Ak4 := GroupRing( GF(5), DihedralGroup(4) ); gap> Size( Ak4 ); @@ -136,8 +133,7 @@ Crossed module [I(GF5[k4])->GF5[k4]] :- 0)*f2 ] gap> Size2d( XIAk4 ); [ 125, 625 ] -]]> - +]]> - - RepresentationsOfObject( XIAk4 ); [ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModAlgebraObj" ] gap> KnownPropertiesOfObject( XIAk4 ); @@ -204,8 +199,7 @@ gap> KnownPropertiesOfObject( XIAk4 ); gap> KnownAttributesOfObject( XIAk4 ); [ "Name", "LeftActingDomain", "Range", "Source", "Boundary", "Size2d", "XModAlgebraAction" ] -]]> - +]]> M. - - XAn := XModAlgebraByMultiplierAlgebra( An ); [ An -> ] gap> XModAlgebraAction( XAn ); IdentityMapping( ) -]]> - +]]> - - X2 := XModAlgebraBySurjection( nat2 );; gap> Display( X2 ); Crossed module [A2->Q2] :- @@ -255,8 +246,7 @@ Crossed module [A2->Q2] :- [ v.1, v.2 ] : Boundary homomorphism maps source generators to: [ v.1 ] -]]> - +]]> <#Include Label="XModAlgebraByBoundaryAndAction"> @@ -283,8 +273,7 @@ of a given crossed module. - - e4 := Elements( IAk4 )[4]; (Z(5)^0)* of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 gap> Je4 := Ideal( IAk4, [e4] );; @@ -304,8 +293,7 @@ Crossed module [ -> ..] :- [ (Z(5)^0)* of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ] : Boundary homomorphism maps source generators to: [ (Z(5)^0)* of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ] -]]> - +]]> @@ -381,8 +369,7 @@ which may have the attributes listed. - - c4 := CyclicGroup( 4 );; gap> Ac4 := GroupRing( GF(2), c4 ); @@ -425,8 +412,7 @@ gap> IsTotal( mor ); true gap> IsSingleValued( mor ); true -]]> - +]]> @@ -447,8 +433,7 @@ An example is given below. - - Xmor := Kernel( mor ); [ -> ] gap> IsXModAlgebra( Xmor ); @@ -457,8 +442,7 @@ gap> Size2d( Xmor ); [ 4, 4 ] gap> IsSubXModAlgebra( XIAc4, Xmor ); true -]]> - +]]> \theta and \varphi. - - ic4 := One( Ac4 );; gap> e1 := ic4*c4.1 + ic4*c4.2; (Z(2)^0)*f1+(Z(2)^0)*f2 @@ -535,8 +518,7 @@ Crossed module [..->..] :- : Boundary homomorphism maps source generators to: [ (Z(2)^0)* of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] -]]> - +]]>