Journal of Symbolic Logic 60 (3):843-860 (1995)

We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming "Swiss cheeses" in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in "most" valued fields F, if f(x),g(x) ∈ F[ x] and v is the valuation map, then the set {x : v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of "valued trees", which we define formally
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DOI 10.2307/2275760
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Real Closed Rings II. Model Theory.Gregory Cherlin & Max A. Dickmann - 1983 - Annals of Pure and Applied Logic 25 (3):213-231.
Complete Theories.Robert L. Vaught - 1960 - Journal of Symbolic Logic 25 (2):172-174.
Cell Decompositions of C-Minimal Structures.Deirdre Haskell & Dugald Macpherson - 1994 - Annals of Pure and Applied Logic 66 (2):113-162.

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Imaginaries in Real Closed Valued Fields.Timothy Mellor - 2006 - Annals of Pure and Applied Logic 139 (1):230-279.
Grothendieck Rings of Theories of Modules.Amit Kuber - 2015 - Annals of Pure and Applied Logic 166 (3):369-407.
A criterion for uniform finiteness in the imaginary sorts.Will Johnson - 2022 - Archive for Mathematical Logic 61 (3):583-589.

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