Locally Constant Functions in C-Minimal Structures

نویسنده

  • Pablo Cubides Kovacsics
چکیده

Let M be a C-minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → T in C-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a description of definable subsets of T in one variable and analogues of known results in algebraically closed valued fields. This paper studies the behavior of definable locally constant functions f : M → T where M is a C-minimal structure, T is its canonical tree, each node of T has infinitely many branches and branches are densely ordered. These conditions are fulfilled in different C-minimal ultrametric spaces such as algebraically closed valued fields, where T corresponds to the set of closed balls not having ∞ as their radius (i.e., not being a singleton) ordered by inclusion. Locally constant functions appear naturally when one gives a general description of definable sets in C-minimal structures such as the cell decomposition theorem proved by Haskell and Macpherson in [4], especially if the theory satisfies the exchange property. In particular, we show that (all terms to be later defined): Theorem. Let f : M → T be a partial definable locally constant function. Then dom(f) can be decomposed (possibly adding parameters) into cells D1, . . . , Dn such that for each 1 ≤ i ≤ n one and only one of the following conditions holds: 1. f(Di) is an antichain. 2. f(Di) is a chain. We use the result to show that the domain of a partial definable function from M to a branch B of T can be decomposed into finitely many cells D1, . . . , Dn Printed July 9, 2013

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عنوان ژورنال:
  • J. Symb. Log.

دوره 80  شماره 

صفحات  -

تاریخ انتشار 2015