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Given f ∈ C[[z Z ]] , let {z1 1, j zn n, j : 1 j p} be the set of dominant monomials of f . Then we may take αi = min1 j p βi,j for each i. Often, we rather assume that z 1 is an infinitely large variable. In that case, z A is given the opposite ordering z α ≺ z β ⇔ α < β. 4. There are two ways of explicitly forming rings of multivariate grid-based series: let z1, , zn be formal variables and A1, , An ordered additive monoids. Then we define the rings of natural grid-based power series resp. recursive grid-based power series in z1, , zn over C and along A1, , An by C[[z1A1, C[[z1A1; If A1 = , znAn]] = C[[z1A1 × ; znAn]] = C[[z1A1 × znAn]]; znAn]].

Let A be a totally ordered R-algebra. We may totally order the polynomial extension A[ε] of A by an infinitesimal element ε by setting a0 + a1 ε + + ad εd > 0, if and only if there exists an index i with a0 = = . Similarly, ai−1 = 0 < ai. This algebra is non-archimedean, since 1 ε ε2 one may construct an extension A[ω] with an infinitely large element ω, in which 1 ≺ ω ≺ ω 2 ≺ . 21. a) Given a totally ordered vector space V over a totally ordered field K, show that x y ⇔ ∃λ ∈ K , |x| λ y; x ≺ y ⇔ ∀λ ∈ K , λ x < |y |.

We will denote families by calligraphic characters F , G , and write F (S) for the collection of all families with values in S. Explicit families (fi)i∈I will sometimes be denoted by (fi: i ∈ I). Consider two families F = (fi)i∈I ∈ S I and G = (g j ) ∈ S J , where I, J and S are arbitrary sets. Then we define F G = (hi)i∈I J, where hi = fi if i ∈ I gi if i ∈ J F × G = (fi , gj )(i, j)∈I ×J More generally, if I = j ∈J Ij , and G j = (fi)i∈I j for all j ∈ J , then we denote Gj = F . 4 Strong linear algebra Given an operation ϕ: S1 × k = 1, , n, we define ϕ(F1, 45 × Sn → T and families Fk = (fk,i)i∈Ik ∈ SkIk for , Fn) = (ϕ(f1,i1, , fn,in))(i1, ,in)∈I1 × ×In.