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Cell decompositions of $$C$$-minimal structures. (English) Zbl 0790.03039
$$C$$-minimality is a variant of $$o$$-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on a set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for $$C$$-minimal structures is proved, and a notion of dimension is introduced. It is shown that $$C$$- minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has the exchange property, and for definable sets dimension coincides with the rank obtained from algebraic closure.
Reviewer: D.Macpherson

##### MSC:
 03C60 Model-theoretic algebra 12L12 Model theory of fields 03C45 Classification theory, stability, and related concepts in model theory 12J10 Valued fields
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