Totally ordered fields with symmetric gaps

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 е A there exists a₁ е A such that (a₁ + (a₁ - a)) е 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) = p > X₀ , where G is the group of Archimedean classes of K and cofinality of each symmetric gap of K is p. We show that K is order-isomorphic to the field of bounded formal power series ^[[G,p]]. We prove that a gap (A,B) of an ordered field K is symmetric iff 3t е ^[[G]] \ K , A < t < B , where G is the group of Archimedean classes of K . For any ordered field, with a symmetric gap of cofinality p we construct a subfield, with a symmetric gap of the same cofinality. We consider an example of real closed field H, tf[[G,P]] с H с tf[[G,p+ ]], with a symmetric gap of cofinality p⁺ .

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