Cell decompositions of C-minimal structures

Haskell, Deirdre; Macpherson, Dugald

doi:10.1016/0168-0072(94)90064-7

Cited by 56 publications

(89 citation statements)

References 12 publications

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“…, b n }/A) = r. It is an easy exercise to show that rk(X) does not depend on the choice of A, a, and φ. We prove the following theorem, which has analogues for o-minimal and C-minimal structures (see Proposition 6.3 of [12] for the latter).…”

Section: Suppose First That π(X) Is An Open Cell In Mmentioning

confidence: 99%

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Weakly o-minimal structures and real closed fields

Macpherson

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Steinhorn

2000

Trans. Amer. Math. Soc.

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Abstract. A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

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Section: Suppose First That π(X) Is An Open Cell In Mmentioning

confidence: 99%

“…The final example does not exhibit any pathology, but makes a connection with combinatorial work of Cameron [4], and also with the notion of C-minimality studied in [19] and [12]. Example 2.6.6.…”

Section: 5mentioning

confidence: 99%

Weakly o-minimal structures and real closed fields

Macpherson

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Steinhorn

2000

Trans. Amer. Math. Soc.

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136

105

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“…Also cell decomposition for C-minimal structures [11] is somehow complicated. In v-minimality [13], cell decomposition appears mainly implicitely.…”

Section: Cell Decompositionmentioning

confidence: 99%

An introduction to b-minimality

Cluckers¹

2009

Logic Colloquium 2006

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“…Conversely, any C-field can be made into a valued field such that the C-relation and the valuation satisfy the relation above. It was shown in [5] and [1] that the C-minimal C-fields correspond to the algebraically closed valued fields. With the induced C-relation, the additive group and the multiplicative group of a C-minimal C-field F are C-minimal groups.…”

Section: Uzv))mentioning

confidence: 99%

“…Less developed than o-minimality for the moment, this notion has already led to some promising results (see [5] and [1]). It applies to expansions of algebraically closed valued fields ( [4]), and may be expected to have a development in some ways analogous to o-minimality (see [1]). Some of the tools of stability can be developed in this context ( [2], [3]).…”

Section: Introductionmentioning

confidence: 99%