Some examples of valued fields (Q1891762)

The author uses a generalized power series \(A= [[ R^{S,\leq} ]]\) to provide two classes of examples of valued fields, where \(A\) is the set of all mappings \(f: S\to R\) (\(S\) is a torsion-free abelian group with a compatible partial order \(\leq\)) with artinian and narrow support. A lot of results were proved by the author in [Manuscr. Math. 75, 115-150 (1992; Zbl 0767.12001)] which are of the following character: If the field \(R\) has a property \((P)\) and a torsion-free subtotally ordered group \((S, \leq)\) has a property \((Q)\), then \(A\) has also the property \((P)\). In this paper in the first set of examples the valued fields are henselian in general not maximal, with given residue field and value group and in the second set of examples, the valued fields are all maximal and their families of residue fields and value groups form chains. Some additional properties of these examples are investigated.