xmod.tst

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#W  xmod.tst, 14/02/15          XModAlg test files      Z. Arvasi - A. Odabas      
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gap> saved_infolevel_xmodalg := InfoLevel( InfoXModAlg );; 
gap> SetInfoLevel( InfoXModAlg, 0 );


## Chapter 2,  Section 2.1.2
gap> A := GroupRing(GF(5),DihedralGroup(4));
<algebra-with-one over GF(5), with 2 generators>
gap> Size(A);
625
gap> SetName(A,"GF5[D4]");
gap> I := AugmentationIdeal(A);
<two-sided ideal in GF5[D4], (2 generators)>
gap> Size(I);
125
gap> SetName(I,"Aug");
gap> CM := XModAlgebraByIdeal(A,I);
[Aug->GF5[D4]]
gap> Display(CM);

Crossed module [Aug->GF5[D4]] :- 
: Source algebra Aug has generators:
  [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...+(Z(5)^0)*f2 ]
: Range algebra GF5[D4] has generators:
  [ (Z(5)^0)*<identity> of ..., (Z(5)^0)*f1, (Z(5)^0)*f2 ]
: Boundary homomorphism maps source generators to:
  [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...+(Z(5)^0)*f2 ]

gap> Size(CM);
[ 125, 625 ]
gap> f := Boundary(CM);
MappingByFunction( Aug, GF5[D4], function( i ) ... end )
gap> Print( RepresentationsOfObject(CM), "\n" ); 
[ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModAlgebraObj" ]
gap> props := [ "CanEasilyCompareElements", "CanEasilySortElements", 
>  "IsDuplicateFree", "IsLeftActedOnByDivisionRing", "IsAdditivelyCommutative", 
>  "IsLDistributive", "IsRDistributive", "IsPreXModDomain", "Is2dAlgebraObject", 
>  "IsPreXModAlgebra", "IsXModAlgebra" ];;
gap> known := KnownPropertiesOfObject(CM);;
gap> ForAll( props, p -> (p in known) );
true 
gap> Print( KnownAttributesOfObject(CM), "\n" ); 
[ "Name", "Size", "Range", "Source", "Boundary", "XModAlgebraAction" ]


## Chapter 2,  Section 2.1.3
gap> e4 := Elements(I)[4];
(Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2
gap> J := Ideal( I, [e4] );
<two-sided ideal in Aug, (1 generators)>
gap> Size(J);
5
gap> SetName( J, "<e4>" ); 
gap> PM := XModAlgebraByIdeal( A, J );
[<e4>->GF5[D4]]
gap> Display( PM );        

Crossed module [<e4>->GF5[D4]] :- 
: Source algebra <e4> has generators:
  [ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ]
: Range algebra GF5[D4] has generators:
  [ (Z(5)^0)*<identity> of ..., (Z(5)^0)*f1, (Z(5)^0)*f2 ]
: Boundary homomorphism maps source generators to:
  [ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ]

gap> IsSubXModAlgebra( CM, PM );
true


## Chapter 2,  Section 2.1.4
gap> G := SmallGroup(4,2);
<pc group of size 4 with 2 generators>
gap> F := GaloisField(4);
GF(2^2)
gap> R := GroupRing( F, G );
<algebra-with-one over GF(2^2), with 2 generators>
gap> Size(R);
256
gap> SetName( R, "GF(2^2)[k4]" ); 
gap> e5 := Elements(R)[5]; 
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> S := Subalgebra( R, [e5] ); 
<algebra over GF(2^2), with 1 generators>
gap> SetName( S, "<e5>" );
gap> RS := Cartesian( R, S );; 
gap> SetName( RS, "GF(2^2)[k4] x <e5>" ); 
gap> act := AlgebraAction( R, RS, S );;
gap> bdy := AlgebraHomomorphismByFunction( S, R, r->r );
MappingByFunction( <e5>, GF(2^2)[k4], function( r ) ... end )
gap> IsAlgebraAction( act ); 
true
gap> IsAlgebraHomomorphism( bdy );
true
gap> XM := PreXModAlgebraByBoundaryAndAction( bdy, act );
[<e5>->GF(2^2)[k4]]
gap> IsXModAlgebra( XM );
true
gap> Display( XM );

Crossed module [<e5>->GF(2^2)[k4]] :-
: Source algebra has generators:
  [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]
: Range algebra GF(2^2)[k4] has generators:
  [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f2 ]
: Boundary homomorphism maps source generators to:
  [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]

## Chapter 2,  Section 2.2.1
  
gap> A:=GroupRing(GF(2),CyclicGroup(4));
<algebra-with-one over GF(2), with 2 generators>
gap> B:=AugmentationIdeal(A);
<two-sided ideal in <algebra-with-one over GF(2), with 2 generators>, (dimension 3)>
gap> X1:=XModAlgebra(A,B);
[Algebra( GF(2), [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f2+(Z(2)^0)*f1*f2
 ] )->AlgebraWithOne( GF(2), [ (Z(2)^0)*f1, (Z(2)^0)*f2 ] )]
gap> C:=GroupRing(GF(2),SmallGroup(4,2));
<algebra-with-one over GF(2), with 2 generators>
gap> D:=AugmentationIdeal(C);
<two-sided ideal in <algebra-with-one over GF(2), with 2 generators>, (dimension 3)>
gap> X2:=XModAlgebra(C,D);
[Algebra( GF(2), [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f2+(Z(2)^0)*f1*f2
 ] )->AlgebraWithOne( GF(2), [ (Z(2)^0)*f1, (Z(2)^0)*f2 ] )]
gap> B = D;
false
gap> all_f := AllHomsOfAlgebras(A,C);;
gap> all_g := AllHomsOfAlgebras(B,D);;
gap> mor := XModAlgebraMorphism(X1,X2,all_g[1],all_f[2]);
[[..] => [..]]
gap> Display(mor);

Morphism of crossed modules :-
: Source = [Algebra( GF(2), [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2,
  (Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] )->AlgebraWithOne( GF(2), [ (Z(2)^0)*f1, (Z(2)^0)*f2 ] )] with generating sets:
  [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]
  [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f2 ]
:  Range = [Algebra( GF(2), [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2,
  (Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] )->AlgebraWithOne( GF(2), [ (Z(2)^0)*f1, (Z(2)^0)*f2 ] )] with generating sets:
  [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f2+(Z(2)^0)*f1*f2 ]
  [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1, (Z(2)^0)*f2 ]
: Source Homomorphism maps source generators to:
  [ <zero> of ..., <zero> of ..., <zero> of ... ]
: Range Homomorphism maps range generators to:
  [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*<identity> of ..., (Z(2)^0)*<identity> of ... ]

gap> X3 := Kernel(mor);
[Algebra( GF(2), [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f2+(Z(2)^0)*f1*f2
 ] )->Algebra( GF(2), [ (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f1*f2, (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1
 ] )]
gap> IsTotal(mor);
true
gap> IsSingleValued(mor);
true

## Chapter 2,  Section 2.2.2

gap> IsXModAlgebra(X3);
true
gap> Size(X3);
[ 8, 8 ]
gap> IsSubXModAlgebra(X1,X3);
true

## Chapter 2,  Section 2.2.3

gap> theta := SourceHom(mor);
[ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*f1+(Z(2)^0)*f2, (Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] ->
[ <zero> of ..., <zero> of ..., <zero> of ... ]
gap> phi := RangeHom(mor);
[ (Z(2)^0)*f1 ] -> [ (Z(2)^0)*<identity> of ... ]
gap> IsInjective(mor);
false
gap> IsSurjective(mor);
false



  
 


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#E  xmod.tst . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here