StandardBasis( M )
StandardBasis( M, seedvectors )
returns the standard basis of the row module M with respect to the seed vectors in the list seedvectors. If no second argument is given the generators of M are taken.
The standard basis is defined as follows. Take the first seed vector v, apply the generators of the ring R acting on M in turn, and if the image is linearly independent of the basis vectors found up to this time, it is added to the basis. When the space becomes stable under the action of R, proceed with the next seed vector, and so on.
Note that you do not get a basis of the whole module if all seed vectors lie in a proper submodule.
gap> s:= Submodule( nat, [ [ 1, 1, 0 ] * Z(2) ] );
Submodule( nat, [ [ Z(2)^0, Z(2)^0, 0*Z(2) ] ] )
gap> b:= StandardBasis( s );
StandardBasis( Submodule( nat, [ [ Z(2)^0, Z(2)^0, 0*Z(2) ] ] ) )
gap> b.vectors;
[ [ Z(2)^0, Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0, Z(2)^0 ] ]
gap> StandardBasis( s, [ [ 0, 1, 1 ] * Z(2) ] );
StandardBasis( Submodule( nat, [ [ Z(2)^0, Z(2)^0, 0*Z(2) ] ] ),
[ [ 0*Z(2), Z(2)^0, Z(2)^0 ], [ Z(2)^0, 0*Z(2), Z(2)^0 ] ] )
GAP 3.4.4