Structure theorems in tame expansions of o-minimal structures by a dense set
Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ and p...
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| Autor*in: |
Eleftheriou, Pantelis E. [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2020 |
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| Anmerkung: |
© The Hebrew University of Jerusalem 2020 |
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| Übergeordnetes Werk: |
Enthalten in: Israel journal of mathematics - The Hebrew University Magnes Press, 1963, 239(2020), 1 vom: Aug., Seite 435-500 |
|---|---|
| Übergeordnetes Werk: |
volume:239 ; year:2020 ; number:1 ; month:08 ; pages:435-500 |
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| DOI / URN: |
10.1007/s11856-020-2058-0 |
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| 520 | |a Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. | ||
| 700 | 1 | |a Günaydin, Ayhan |4 aut | |
| 700 | 1 | |a Hieronymi, Philipp |4 aut | |
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10.1007/s11856-020-2058-0 doi (DE-627)OLC2119880689 (DE-He213)s11856-020-2058-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Eleftheriou, Pantelis E. verfasserin aut Structure theorems in tame expansions of o-minimal structures by a dense set 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Hebrew University of Jerusalem 2020 Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. Günaydin, Ayhan aut Hieronymi, Philipp aut Enthalten in Israel journal of mathematics The Hebrew University Magnes Press, 1963 239(2020), 1 vom: Aug., Seite 435-500 (DE-627)129552283 (DE-600)219689-X (DE-576)015007081 0021-2172 nnns volume:239 year:2020 number:1 month:08 pages:435-500 https://doi.org/10.1007/s11856-020-2058-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-JFK GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4311 AR 239 2020 1 08 435-500 |
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10.1007/s11856-020-2058-0 doi (DE-627)OLC2119880689 (DE-He213)s11856-020-2058-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Eleftheriou, Pantelis E. verfasserin aut Structure theorems in tame expansions of o-minimal structures by a dense set 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Hebrew University of Jerusalem 2020 Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. Günaydin, Ayhan aut Hieronymi, Philipp aut Enthalten in Israel journal of mathematics The Hebrew University Magnes Press, 1963 239(2020), 1 vom: Aug., Seite 435-500 (DE-627)129552283 (DE-600)219689-X (DE-576)015007081 0021-2172 nnns volume:239 year:2020 number:1 month:08 pages:435-500 https://doi.org/10.1007/s11856-020-2058-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-JFK GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4311 AR 239 2020 1 08 435-500 |
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10.1007/s11856-020-2058-0 doi (DE-627)OLC2119880689 (DE-He213)s11856-020-2058-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Eleftheriou, Pantelis E. verfasserin aut Structure theorems in tame expansions of o-minimal structures by a dense set 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Hebrew University of Jerusalem 2020 Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. Günaydin, Ayhan aut Hieronymi, Philipp aut Enthalten in Israel journal of mathematics The Hebrew University Magnes Press, 1963 239(2020), 1 vom: Aug., Seite 435-500 (DE-627)129552283 (DE-600)219689-X (DE-576)015007081 0021-2172 nnns volume:239 year:2020 number:1 month:08 pages:435-500 https://doi.org/10.1007/s11856-020-2058-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-JFK GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4311 AR 239 2020 1 08 435-500 |
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10.1007/s11856-020-2058-0 doi (DE-627)OLC2119880689 (DE-He213)s11856-020-2058-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Eleftheriou, Pantelis E. verfasserin aut Structure theorems in tame expansions of o-minimal structures by a dense set 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Hebrew University of Jerusalem 2020 Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. Günaydin, Ayhan aut Hieronymi, Philipp aut Enthalten in Israel journal of mathematics The Hebrew University Magnes Press, 1963 239(2020), 1 vom: Aug., Seite 435-500 (DE-627)129552283 (DE-600)219689-X (DE-576)015007081 0021-2172 nnns volume:239 year:2020 number:1 month:08 pages:435-500 https://doi.org/10.1007/s11856-020-2058-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-JFK GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4311 AR 239 2020 1 08 435-500 |
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10.1007/s11856-020-2058-0 doi (DE-627)OLC2119880689 (DE-He213)s11856-020-2058-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Eleftheriou, Pantelis E. verfasserin aut Structure theorems in tame expansions of o-minimal structures by a dense set 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Hebrew University of Jerusalem 2020 Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. Günaydin, Ayhan aut Hieronymi, Philipp aut Enthalten in Israel journal of mathematics The Hebrew University Magnes Press, 1963 239(2020), 1 vom: Aug., Seite 435-500 (DE-627)129552283 (DE-600)219689-X (DE-576)015007081 0021-2172 nnns volume:239 year:2020 number:1 month:08 pages:435-500 https://doi.org/10.1007/s11856-020-2058-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT SSG-OPC-JFK GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4311 AR 239 2020 1 08 435-500 |
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Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. © The Hebrew University of Jerusalem 2020 |
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Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. © The Hebrew University of Jerusalem 2020 |
| abstract_unstemmed |
Abstract We study sets and groups definable in tame expansions of o-minimal structures. Let $$\widetilde{\cal M} = \left\langle {{\cal M},P} \right\rangle $$ be an expansion of an o-minimal $${\cal L}$$-structure ℳ by a dense set P. We impose three tameness conditions on $$\widetilde{\cal M}$$ 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 $$\widetilde{\cal M}$$, 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 $${\cal L}$$-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 $${\cal L}$$-definable map. © The Hebrew University of Jerusalem 2020 |
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Structure theorems in tame expansions of o-minimal structures by a dense set |
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Günaydin, Ayhan Hieronymi, Philipp |
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