Symmetry of Sections in Fields of Formal Power Series and a Non-Standard Real Line

Let $R[[G,\beta]]$ be a field of formal power series with real coefficients, whose supports are well ordered subsets of an Abelian group $G$ of cardinality strictly less than $\beta$. For $R[[G,\beta]]$, we give criteria of a section being symmetric and of a symmetric section being Dedekind. It is proved that an $\alpha^+$-saturated non-standard real line $^{*}R$ is isomorphic to some field of the form $R[[G,\alpha^+]]$. For $^{*}R$, some consequences are inferred regarding symmetric sections, and the cofinality of “banks” of the sections.