By Kreuzer and Robbiano

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Then an inequality γ1 ◦ γ3 ≥σ γ2 ◦ γ3 implies γ1 ≥σ γ2 .

N ) is 1 · · · xn clearly an isomorphism of monoids. The monoideal (log(t1 ), log(t2 ), . ) ⊆ Nn is finitely generated by the previous proposition. Thus there exists a number N > 0 such that this monoideal is generated by {log(t1 ), . . , log(tN )}. Consequently, the monoideal (t1 , t2 , . ) ⊆ Tn is generated by {t1 , . . , tN } . As we shall see, ideals and modules generated by terms have many special properties. We begin our studies by giving them a special name. Let R be a ring, let n ≥ 1 , let P = R[x1 , .

4 Term Orderings Dura Lex, sed Lex. 1 where we discussed the notion of polynomial rings in one and several indeterminates. A univariate polynomial with coefficients in a ring R is an expression of the type f (x) = ri xi . One question is: in how many different ways can we write f (x)? We might agree that the coefficients should be written before the corresponding power product and also decide to be “nice” and avoid the + sign before the first coefficient, but still we have to face the commutative property of the sum, which implies that for instance 1 + 2x − 3x2 can also be written as 1 − 3x2 + 2x .

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Computational Commutative Algebra by Kreuzer and Robbiano

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