Truncation in Hahn Fields

Given such a series a = ∑ γ aγt γ ∈ K and α ∈ Γ we can truncate the series at t to give the series a|α := ∑ γ<α aγt γ ∈ K. We call a|α a truncation of a; if in addition a 6= a|α (equivalently, α 6 β for some β ∈ supp a), then a|α is said to be a proper truncation of a. Thus all truncations of a are proper iff supp a is cofinal in Γ. The only proper truncation of ct with nonzero c ∈ k is 0. A subset of K is said to be closed under truncation (or just truncation closed) if it contains all truncations of all its elements. The property of being truncation closed turns out to be stable under various operations. I learned about this from Ressayre [10] (where k = R) and confess to being very surprised by this phenomenon. My modest goal here is to bring together in one place the various stability results of this kind, with proofs and without unnecessary restrictions. For example, the next theorem (from Section 2) gives the stability of truncation closedness under certain arithmetic extension procedures.