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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="UTF-8"?>

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<!-- This is an automatically generated file. -->

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<Chapter Label="Chapter_Examples_and_Tests">

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<Heading>Examples and Tests</Heading>

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<Section Label="Chapter_Examples_and_Tests_Section_Basic_Commands">

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<Heading>Basic Commands</Heading>

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<Example><![CDATA[

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gap> Q := HomalgFieldOfRationals();;

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gap> a := VectorSpaceObject( 3, Q );

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<A vector space object over Q of dimension 3>

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gap> b := VectorSpaceObject( 4, Q );

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<A vector space object over Q of dimension 4>

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gap> homalg_matrix := HomalgMatrix( [ [ 1, 0, 0, 0 ],

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>                                   [ 0, 1, 0, -1 ],

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>                                   [ -1, 0, 2, 1 ] ], 3, 4, Q );

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<A 3 x 4 matrix over an internal ring>

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gap> alpha := VectorSpaceMorphism( a, homalg_matrix, b );

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<A morphism in Category of matrices over Q>

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gap> Display( alpha );

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[ [   1,   0,   0,   0 ],

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  [   0,   1,   0,  -1 ],

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  [  -1,   0,   2,   1 ] ]

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A morphism in Category of matrices over Q

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gap> homalg_matrix := HomalgMatrix( [ [ 1, 1, 0, 0 ],

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>                                   [ 0, 1, 0, -1 ],

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>                                   [ -1, 0, 2, 1 ] ], 3, 4, Q );

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<A 3 x 4 matrix over an internal ring>

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gap> beta := VectorSpaceMorphism( a, homalg_matrix, b );

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<A morphism in Category of matrices over Q>

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gap> CokernelObject( alpha );

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<A vector space object over Q of dimension 1>

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gap> c := CokernelProjection( alpha );;

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gap> Display( c );

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[ [     0 ],

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  [     1 ],

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  [  -1/2 ],

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  [     1 ] ]

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A split epimorphism in Category of matrices over Q

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gap> gamma := UniversalMorphismIntoDirectSum( [ c, c ] );;

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gap> Display( gamma );

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[ [     0,     0 ],

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  [     1,     1 ],

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  [  -1/2,  -1/2 ],

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  [     1,     1 ] ]

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A morphism in Category of matrices over Q

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gap> colift := CokernelColift( alpha, gamma );;

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gap> IsEqualForMorphisms( PreCompose( c, colift ), gamma );

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true

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gap> FiberProduct( alpha, beta );

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<A vector space object over Q of dimension 2>

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gap> F := FiberProduct( alpha, beta );

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<A vector space object over Q of dimension 2>

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gap> p1 := ProjectionInFactorOfFiberProduct( [ alpha, beta ], 1 );

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<A morphism in Category of matrices over Q>

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gap> Display( PreCompose( p1, alpha ) );

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[ [   0,   1,   0,  -1 ],

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  [  -1,   0,   2,   1 ] ]

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A morphism in Category of matrices over Q

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gap> Pushout( alpha, beta );

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<A vector space object over Q of dimension 5>

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gap> i1 := InjectionOfCofactorOfPushout( [ alpha, beta ], 1 );

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<A morphism in Category of matrices over Q>

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gap> i2 := InjectionOfCofactorOfPushout( [ alpha, beta ], 2 );

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<A morphism in Category of matrices over Q>

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gap> u := UniversalMorphismFromDirectSum( [ b, b ], [ i1, i2 ] );

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<A morphism in Category of matrices over Q>

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gap> Display( u );

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[ [     0,     1,     1,     0,     0 ],

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  [     1,     0,     1,     0,    -1 ],

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  [  -1/2,     0,   1/2,     1,   1/2 ],

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  [     1,     0,     0,     0,     0 ],

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  [     0,     1,     0,     0,     0 ],

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  [     0,     0,     1,     0,     0 ],

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  [     0,     0,     0,     1,     0 ],

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  [     0,     0,     0,     0,     1 ] ]

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A morphism in Category of matrices over Q

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gap> KernelObjectFunctorial( u, IdentityMorphism( Source( u ) ), u ) = IdentityMorphism( VectorSpaceObject( 3, Q ) );

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true

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gap> IsZero( CokernelObjectFunctorial( u, IdentityMorphism( Range( u ) ), u ) );

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true

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gap> DirectProductFunctorial( [ u, u ] ) = DirectSumFunctorial( [ u, u ] );

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true

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gap> CoproductFunctorial( [ u, u ] ) = DirectSumFunctorial( [ u, u ] );

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true

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gap> IsOne( FiberProductFunctorial( [ [ u, IdentityMorphism( Source( u ) ), u ], [ u, IdentityMorphism( Source( u ) ) , u ] ] ) );

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true

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gap> IsOne( PushoutFunctorial( [ [ u, IdentityMorphism( Range( u ) ), u ], [ u, IdentityMorphism( Range( u ) ) , u ] ] ) );

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true

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]]></Example>

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</Section>

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</Chapter>

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