Structure theorems in tame expansions of o-minimal structures by a dense set
Structure theorems in tame expansions of o-minimal structures by a dense set
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o-最小结构通过稠密集进行驯服展开的结构定理
DOI：
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发表时间：
2015
影响因子：
1
通讯作者：
Philipp Hieronymi
中科院分区：
文献类型：
--
作者：
Pantelis E. Eleftheriou;Ayhan Günaydin;Philipp Hieronymi
We study sets and groups definable in tame expansions of o-minimal structures. Let ℳ˜=〈ℳ,P〉documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{cal M} = leftlangle {{cal M},P}
ight
angle $$end{document} be an expansion of an o-minimal ℒdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal L}$$end{document}-structure ℳ by a dense set P. We impose three tameness conditions on ℳ˜documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{cal M}$$end{document} and prove a structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such ℳ˜documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{cal M}$$end{document}, as it achieves a decomposition of definable sets into unions of ‘cones’, instead of only boolean combinations of them. The proofs involve induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an ℒdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal L}$$end{document}-definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an ℒdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal L}$$end{document}-definable map.
We study sets and groups definable in tame expansions of o-minimal structures. Let ℳ˜=〈ℳ,P〉documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{cal M} = leftlangle {{cal M},P}
ight
angle $$end{document} be an expansion of an o-minimal ℒdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal L}$$end{document}-structure ℳ by a dense set P. We impose three tameness conditions on ℳ˜documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{cal M}$$end{document} and prove a structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such ℳ˜documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$widetilde{cal M}$$end{document}, as it achieves a decomposition of definable sets into unions of ‘cones’, instead of only boolean combinations of them. The proofs involve induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an ℒdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal L}$$end{document}-definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an ℒdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${cal L}$$end{document}-definable map.
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