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MATHEMATICS
Totally ordered fields with symmetric gaps
N. Yu. Galanova Tomsk State University, Tomsk, Russian Federation
Abstract:
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
$\exists t\in \mathbf{R}[[G]]\setminus K$,
$A<t<B$, 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
$\beta^+$.
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.
UDC:
512.623.23
Received: 30.08.2016
DOI:
10.17223/19988621/46/2
Fulltext:
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