caa a22 4500 388039213 CHVBK 20180307125016.0 cr unu---uuuuu 161130e199806 xx s 000 0 eng 10.2307/2586860 doi S0022481200015188 pii (NATIONALLICENCE)cambridge-10.2307/2586860 A note on valuation definable expansions of fields [Elektronische Daten] In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters. Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset of F n definable in ℱ v . Then either A is definable in ℱ = (F, +, ×) or V is definable in . Theorem B. Let ℛ v = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset of R n definable in ℛ v . Then either A is definable in ℛ = (R, <, +, ×) or V is definable in . The proofs of Theorems A and B are quite similar. Both ℱ v and ℛ v admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper. Copyright © Association for Symbolic Logic 1998 Haskell Deirdre Mathematics Department, College of the Holy Cross, Worcester MA 01610, USA, E-mail: haskell@math.holycross.edu Dugald Macpherson, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, E-mail: pmthdm@amsta.leeds.ac.uk The Journal of Symbolic Logic Cambridge University Press 63/2(1998-06), 739-743 0022-4812 63:2<739 1998 63 JSL https://doi.org/10.2307/2586860 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586860 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Haskell Deirdre Mathematics Department, College of the Holy Cross, Worcester MA 01610, USA, E-mail: haskell@math.holycross.edu NATIONALLICENCE 700 1- Dugald Macpherson Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England, E-mail: pmthdm@amsta.leeds.ac.uk NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/2(1998-06), 739-743 0022-4812 63:2<739 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge