A characterization of the Dedekind completion of a totally ordered group of infinite rank
Let \(G\) be a directed partially ordered abelian group and \(X\) a nonempty subset of \(G\), throughout. If \(X\) is bounded from above (below), let \(U(X)\) (respectively, \(L(X)\)) denote the set of all the upper (lower) bounds of \(X\). To each \(X\) bounded from above we can associate \(X^{\#}=L(U(X))\). The sets \( X^{\#}\) serve as a generalization of Dedekind cuts to p.o. sets. The set \( G^{\#}\) of all \(X^{\#}\) for suitable subsets \(X\) of \(G\) can be shown to be a conditionally complete lattice under inclusion and a semigroup under a compatible multiplication defined by \((XY)^{\#}=(X^{\#}Y)^{\#}\) [\textit{L. Fuchs}, Partially ordered algebraic systems. Oxford-London-New York-Paris: Pergamon Press (1963; Zbl 0137.02001)]. Also \(G\) can be embedded in \(G^{\#}\) via the mapping \(a\mapsto (a)^{\#}=L(a)\). \(G^{\#}\) is called the Dedekind completion of \(G\). If \(G\) is a lattice-ordered group it can be regarded as a sublattice of \(G^{\#}\). Finally, \(G^{\#}\) is a group if and only if \(G\) is completely integrally closed. In search of norms of vectors and linear operators in non-Archimedian functional analysis the authors of the paper under review study the Dedekind completion \(\Gamma _{\alpha }^{\#}\) of the totally ordered group \(\Gamma _{\alpha }\) where \(\alpha \) is an ordinal and \(\Gamma _{\alpha }\) is a sort of Hahn product of totally ordered groups \(\{G_{\beta }\}_{\beta <\alpha }\) of rank one. The rank of each \(\Gamma _{\alpha }\) is the order type of \( \alpha\). The first author has discussed \(\Gamma _{\alpha }\) in detail in [\textit{E. Olivos}, Ann. Math. Blaise Pascal 12, No. 1, 79--90 (2005; Zbl 1085.06010)]. For this review, we note that with \(\alpha \) and \(G_{\beta }\) as above \(\Gamma _{\alpha }=\{f:\alpha \rightarrow \bigcup G_{\beta }:f(\beta )\in G_{\beta }\) and supp(\( f \)) is finite\}. The multiplication in \(\Gamma _{\alpha }\) is componentwise and the ordering is antilexicographic. The aim of this study is to provide descriptions of elements of \(\Gamma _{\alpha }^{\#}\) when \(\{G_{\beta }\}_{\beta <\alpha }\) are all complete and to treat the more general case of \(\Gamma _{\alpha }^{\#}\) when \(\{G_{\beta }\}_{\beta <\alpha }\) are not all complete.
scientific article; zbMATH DE number 5619298