We define the rings of example 1.5.3, in the Singular book.
i1 : (n,m) = (2,3);
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i2 : A1 = QQ[x_1..x_n,y_1..y_m,MonomialOrder=>{n, RevLex=>m},Global=>false];
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i3 : f = x_1*x_2^2 + 1 + y_1^10 + x_1*y_2^5 + y_3
2 5 10
o3 = x x + x y + 1 + y + y
1 2 1 2 3 1
o3 : A1
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i4 : 1_A1 > y_1^10
o4 = true
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The second monomial order has the first block local, and the second block polynomial.
i5 : A2 = QQ[x_1..x_n,y_1..y_m,MonomialOrder=>{RevLex=>n, m},Global=>false];
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i6 : substitute(f,A2)
10 5 2
o6 = y + y + 1 + x y + x x
1 3 1 2 1 2
o6 : A2
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i7 : x_1*y_2^5 < 1_A2
o7 = true
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The third example has three blocks of variables.
i8 : A3 = QQ[x_1..x_n,y_1..y_m,MonomialOrder=>{n, RevLex=>2, m-2},Global=>false];
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i9 : substitute(f,A3)
2 5 10
o9 = x x + x y + y + 1 + y
1 2 1 2 3 1
o9 : A3
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