Totally ordered fields with symmetric gaps

N. Yu. Galanova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 46 (2017), pp. 14-20

Résumé

The paper investigates properties of totally ordered fields with symmetric gaps. Let $(A, B)$ be a gap of an ordered field $K$. The set $A$ is called long-shore if for all $a\in A$ there exists $a_1\in A$ such that $(a_1+(a_1-a))\in B$. If both of the shores $A$ and $B$ are long-shore, then the gap $(A, B)$ is called symmetric. We consider under (GCH) a real closed field $K$, $|K|=|G|=cf(G)=\beta>\aleph_0$, where $G$ is the group of Archimedean classes of $K$ and cofinality of each symmetric gap of $K$ is $\beta$. We show that $K$ is order-isomorphic to the field of bounded formal power series $\mathbf{R}[[G, \beta]]$. We prove that a gap $(A, B)$ of an ordered field $K$ is symmetric iff $\exists t\in \mathbf{R}[[G]]\setminus K$, $A$, where $G$ is the group of Archimedean classes of $K$. For any ordered field, with a symmetric gap of cofinality $\beta$ we construct a subfield, with a symmetric gap of the same cofinality. We consider an example of real closed field $H$, $\mathbf{R}[[G, \beta]]\subset H\subset\mathbf{R}[[G, \beta^+]]$, with a symmetric gap of cofinality $\beta^+$.

Détail

Keywords: totally ordered Abelian group, totally ordered field, field of bounded formal power series, simple transcendental extension of ordered field, real closure, symmetric gap, cofinality of a gap.