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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%A module.msk GAP documentation Thomas Breuer
%%
%A @(#)$Id: module.msk,v 1.12 2002/04/15 10:02:30 sal Exp $
%%
%Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
%Y Copyright (C) 2002 The GAP Group
%%
\PreliminaryChapter{Modules}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Generating modules}
\Declaration{IsLeftOperatorAdditiveGroup}
\Declaration{IsLeftModule}
\beginexample
gap> V:= FullRowSpace( Rationals, 3 );
( Rationals^3 )
gap> IsLeftModule( V );
true
\endexample
\Declaration{GeneratorsOfLeftOperatorAdditiveGroup}
\Declaration{GeneratorsOfLeftModule}
\beginexample
gap> V:= FullRowSpace( Rationals, 3 );;
gap> GeneratorsOfLeftModule( V );
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
\endexample
\Declaration{AsLeftModule}
\beginexample
gap> coll:= [ [0*Z(2),0*Z(2)], [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];
[ [ 0*Z(2), 0*Z(2) ], [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ],
[ Z(2)^0, Z(2)^0 ] ]
gap> AsLeftModule( GF(2), coll );
<vector space of dimension 2 over GF(2)>
\endexample
\Declaration{IsRightOperatorAdditiveGroup}
\Declaration{IsRightModule}
\Declaration{GeneratorsOfRightOperatorAdditiveGroup}
\Declaration{GeneratorsOfRightModule}
\Declaration{LeftModuleByGenerators}
\beginexample
gap> coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];;
gap> V:= LeftModuleByGenerators( GF(16), coll );
<vector space over GF(2^4), with 3 generators>
\endexample
\Declaration{LeftActingDomain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Submodules}
\Declaration{Submodule}
\beginexample
gap> coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];;
gap> V:= LeftModuleByGenerators( GF(16), coll );;
gap> W:= Submodule( V, [ coll[1], coll[2] ] );
<vector space over GF(2^4), with 2 generators>
gap> Parent( W ) = V;
true
\endexample
\Declaration{SubmoduleNC}
\Declaration{ClosureLeftModule}
\beginexample
gap> V:= LeftModuleByGenerators( Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] );
<vector space over Rationals, with 2 generators>
gap> ClosureLeftModule( V, [ 1, 1, 1 ] );
<vector space over Rationals, with 3 generators>
\endexample
\Declaration{TrivialSubmodule}
\beginexample
gap> V:= LeftModuleByGenerators( Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ] );;
gap> TrivialSubmodule( V );
<vector space over Rationals, with 0 generators>
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Free Modules}
\Declaration{IsFreeLeftModule}
\Declaration{FreeLeftModule}
\beginexample
gap> V:= FreeLeftModule( Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ], "basis" );
<vector space of dimension 2 over Rationals>
\endexample
%W \Declaration{AsFreeLeftModule}
\Declaration{Dimension}
\beginexample
gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );;
gap> Dimension( V );
2
\endexample
\Declaration{IsFiniteDimensional}
\beginexample
gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );;
gap> IsFiniteDimensional( V );
true
\endexample
\Declaration{UseBasis}
\beginexample
gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );;
gap> UseBasis( V, [ [ 1, 0 ], [ 1, 1 ] ] );
gap> V; # now V knows its dimension
<vector space of dimension 2 over Rationals>
\endexample
\Declaration{IsRowModule}
\Declaration{IsMatrixModule}
\Declaration{IsFullRowModule}
\Declaration{FullRowModule}
\beginexample
gap> V:= FullRowModule( Integers, 5 );
( Integers^5 )
\endexample
\Declaration{IsFullMatrixModule}
\Declaration{FullMatrixModule}
\beginexample
gap> FullMatrixModule( GaussianIntegers, 3, 6 );
( GaussianIntegers^[ 3, 6 ] )
\endexample
\Declaration{IsHandledByNiceBasis}
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