Internal characterizations of lattice-ordered power series fields
The author continues his investigations of lattice-ordered fields [cf. for instance, J. Aust. Math. Soc., Ser. A 52, 299-321 (1992; Zbl 0766.06019)] focusing on problems of construction of lattice-ordered power series fields and of embedding abstract lattice-ordered fields into them. Let \((\Gamma,\cdot,<)\) be a totally ordered abelian group with neutral element \(e\), \({\mathcal A}\) a subset of \(2^\Gamma\), the power set of \(\Gamma\), such that for any \(A\in{\mathcal A}\) all chains in \(A\) contain a maximal element, \(\cup{\mathcal A}\) generates \(\Gamma\), and \({\mathcal A}\) is an ideal in \((2^\Gamma,\subseteq)\) invariant under \(\Gamma\) and is closed with respect to forming the semigroup generated by those \(A\in{\mathcal A}\) which fulfill \(a\leq e\) for all \(a\in A\). Such an \({\mathcal A}\) is called a field family. \textit{F. J. Rayner} [Glasg. Math. J. 9, 146-151 (1968; Zbl 0159.05302)] has shown that any field family \({\mathcal A}\) gives rise to a field structure on the set of all functions \(\Gamma\to T\) with support in \({\mathcal A}\), where \(T\) is chosen to be a totally ordered field. The author generalizes this construction by considering a partial order on \(\Gamma\) such that the positive cone is totally ordered, and several relaxations of the above mentioned axioms of a field family \({\mathcal A}\). He terms these more general subsets \({\mathcal X}\) of \(2^\Gamma\) supporting collections. Here, in general, the set \(F\) of all functions \(\Gamma\to T\) with support in \({\mathcal X}\) can only be given the structure of a lattice-ordered algebra by defining addition coordinatewise and multiplication by a convolution which depends on the (always possible) choice of some factor set. Characterizations are given for the case where \(F\) becomes a field (Proposition 1.1). Let \(L\) be a lattice-ordered field with \(1>0\), and \(M(L)\) denote its maximal totally ordered subfield. In some cases \(L\) admits an embedding into a power series algebra \(F\) of the form described above constructed over \(M(L)\) (Proposition 4.3). In particular (Proposition 5.1), an archimedean lattice-ordered field \(L\) is a power series field iff it has the following property: \(L\) is laterally complete and multiplicatively dense, and the special elements of \(L\) form a group. The case where only \(M(L)\) is archimedean is also considered (Corollary 5.4).
construction of lattice-ordered power series fields
scientific article; zbMATH DE number 926944