in page 137 of an introduction to groebner bases

how to eliminate the redundant solution (y^2-x*z, 0, -x^2+y*w)

from 3 of them?
eliminate({y,y^2-x*z,-z}, {x, y, z, w});
eliminate({-x,0,y}, {x, y, z});
eliminate({w,-x^2+y*w,-x}, {x, y, z, w});

ma := allstructs(Permutation([1, 1, 1, 2, 2, 2, 3, 3, 3]), size = 3);

above is fast

but below is very slow.
ma2 := allstructs(Permutation(ma), size = 3);

just for all combinations of matrix , replicateM in haskell is the fastest.

in maple, ma2 := allstructs(Permutation(ma), size = 3); is very slow

with(Groebner):
T := lexdeg([x,y,z],[e1,e2]);
intermsof1 := y;
intermsof2 := -z;
GB := Basis([e1-intermsof1, e2-intermsof2], 'tord',T);
result := NormalForm(y^2-x*z, GB,'tord', T);
result := NormalForm(y^2-x*z, GB, T);

originally Basis do not have error when without parameter 'tord'

after it has argument error, it has to be added extra parameter tord

NormalForm has the same error too.

i do not understand why it has error, how to solve?

i just want to express y^2-x*z in terms of y and -z

1.

similar gap system, which function can output coset table ?

if find example below,

f := FreeGroup( "a", "b" );

g := f / [f.1^2, f.2^3, (f.1*f.2)^5];
tab := CosetTable(g, Subgroup(g, [g.1, g.2*g.1*g.2*g.1*g.2^-1]));

2.

if quotient group is for equivalence relations, what do relations [g.1, g.2*g.1*g.2*g.1*g.2^-1] for? which kind of relations is for input to subgroup parameter which is like relations?

3. equalivance relations in wiki mentioned is to partition cells, how do they satisfy symmetric, reflexive and transitive after partition? how to test these partition?

how to implement a free group with symmetric properties?

i find that equivalence relations = relexive, symmetric and transitive , is only these three ?

if implement symmetric properties, does it mean that adding equivalence relations to group?

how to implement free group with this?

i guess something like

f := FreeGroup( "a", "b" );
g := f / [equivalence relations....];