# Singular Book 1.2.13 -- monomial orderings

Monomial orderings are specified when defining a polynomial ring.

## global orderings

The default order is the graded (degree) reverse lexicographic order.
 i1 : A2 = QQ[x,y,z]; i2 : A2 = QQ[x,y,z,MonomialOrder=>GRevLex]; i3 : f = x^3*y*z+y^5+z^4+x^3+x*y^2 5 3 4 3 2 o3 = y + x y*z + z + x + x*y o3 : A2
Lexicographic order.
 i4 : A1 = QQ[x,y,z,MonomialOrder=>Lex]; i5 : substitute(f,A1) 3 3 2 5 4 o5 = x y*z + x + x*y + y + z o5 : A1
 i6 : A3 = QQ[x,y,z,MonomialOrder=>{Weights=>{1,1,1},Lex}]; i7 : substitute(f,A3) 3 5 4 3 2 o7 = x y*z + y + z + x + x*y o7 : A3
 i8 : A4 = QQ[x,y,z,MonomialOrder=>{Weights=>{5,3,2},Lex}]; i9 : substitute(f,A4) 3 3 5 2 4 o9 = x y*z + x + y + x*y + z o9 : A4
 i10 : A = QQ[x,y,z,MonomialOrder=>{1,2}]; i11 : substitute(f,A) 3 3 2 5 4 o11 = x y*z + x + x*y + y + z o11 : A
 i12 : A = QQ[x,y,z,MonomialOrder=>{Weights=>{-1,0,0},Weights=>{0,-1,0},Weights=>{0,0,-1}},Global=>false]; i13 : substitute(f,A) 4 5 2 3 3 o13 = z + y + x*y + x + x y*z o13 : A
 i14 : A = QQ[x,y,z,MonomialOrder=>{Weights=>{-1,-1,-1},GRevLex},Global=>false]; i15 : substitute(f,A) 3 2 4 5 3 o15 = x + x*y + z + y + x y*z o15 : A