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# Ordered modules -- Overview of the ordered module $\QQ^n$

Many standard valuations take values in a totally ordered subgroup $\Gamma \subseteq \QQ^n$. These standard valuations implement an instance of the type OrderedQQn, whose order is based on the monomial order of a given ring $R$. The values in $\QQ^n$ are compared using the monomial order of $R$. By default, our valuations use the min convention, that is $v(a + b) \ge \min(v(a), v(b))$.

 i1 : R = QQ[x,y]; i2 : I = ideal(x,y); o2 : Ideal of R i3 : v = leadTermValuation R; i4 : a = v(x) o4 = | -1 | | 0 | o4 : Ordered QQ^2 module i5 : b = v(y) o5 = | 0 | | -1 | o5 : Ordered QQ^2 module i6 : c = v(x+y) o6 = | -1 | | 0 | o6 : Ordered QQ^2 module i7 : a > b o7 = false i8 : a == c o8 = true

Two ordered $\QQ^n$ modules are equal if they are built from the same ring. Note that isomorphic rings with the same term order may not be equal.

 i9 : M1 = orderedQQn(3, {Lex}) 3 o9 = QQ o9 : Ordered QQ^3 module i10 : R = M1.cache.Ring o10 = R o10 : PolynomialRing i11 : M2 = orderedQQn R 3 o11 = QQ o11 : Ordered QQ^3 module i12 : M1 == M2 o12 = true i13 : S = QQ[x_1 .. x_3, MonomialOrder => {Lex}] warning: clearing value of symbol x to allow access to subscripted variables based on it : debug with expression debug 9868 or with command line option --debug 9868 o13 = S o13 : PolynomialRing i14 : M3 = orderedQQn S 3 o14 = QQ o14 : Ordered QQ^3 module i15 : M1 == M3 o15 = false