Description
The option
Eliminate => n is a shortcut for
Weights => {n:1} The remaining variables are given weight 0.The monomial order is the elimination order eliminating the first n variables, refined by the graded reverse lexicographic order.
i1 : R = QQ[a..i, MonomialOrder => Eliminate 3];

This order enables intersections with the subring consisting of all but the first 3 variables. For this, use the command
selectInSubring.
i2 : I = ideal(a^2, bf, d^4, i  b);
o2 : Ideal of R

i3 : selectInSubring(1, gens gb I)
o3 =  fi d4 
1 2
o3 : Matrix R < R

Eliminate may be combined with block orders as well.
i4 : R = QQ[a..i, MonomialOrder => {Eliminate 3,4,2}];

i5 : d^3  a*e^4 + b^2*i + a*c*d*f +a*c^2*g + a*c*g
2 2 4 3
o5 = a*c g + a*c*d*f + b i + a*c*g  a*e + d
o5 : R

In the last example, the order is block order: the first four variables are in the first block, the subsequent two variables are in the second block, the remaining variables are in the third block, and the weights of the variables are 1,1,1,0,..., 0. We illustrate the usage of
selectInSubring.
i6 : I = ideal(a..i)
o6 = ideal (a, b, c, d, e, f, g, h, i)
o6 : Ideal of R

i7 : selectInSubring(1, gens gb I);
1 6
o7 : Matrix R < R

i8 : selectInSubring(2, gens gb I);
1 5
o8 : Matrix R < R

i9 : selectInSubring(3, gens gb I);
1 3
o9 : Matrix R < R
