On cuts of the quotient field of a ring of formal power series

In studies related to the classification of real-closed fields, fields of formal power series with a multiplicative divisible group of Archimedean classes are essentially used. Consider a linearly ordered Abelian divisible group G = G(L,Q), which consists of words with generators from a linearly ordered set L similar to the ordinal ω1 and rational exponents. The article deals with the properties of sections of subfields of the field of bounded formal power series R[[G,ℵ1]]. For all ξi ∈ L we set ti =ξi-1. Consider an infinite strictly decreasing sequence {tγ}γ∈Г, where Г⊆ ω1 \ {1} is an arbitrary infinite set. Series of the form x = ∑rγ tγ ∈ R[[G]], where rγ ≠ 0 for all γ ∈ Г, i.e. supp(х) = {tγ | γ ∈ Г} , we will call series of the form (*). We prove that series of the form (*) for rγ > 0 for all γ ∈ Г generate in the field qfR[[G,ℵ0 ]] = K symmetric non-fundamental sections of confinality (ℵ0,ℵ0), in the real closure qfR[[G,ℵ0 ] ] = K series (*) generate symmetric sections. Let H be the least by inclusion real closed subfield of the field R[[G,ℵ1]] containing K and all truncations of the series x ω1 = ∑ 1•tγ. Then K ≠ H and the elements of the real closure of the simple transcendental extension H(хω1) that do not belong to H generate symmetric sections of the type (ℵ1,ℵ1) in the field H.

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