Alexandrov-Zeeman type theorems expressed in terms of definability

Victor Pambuccian

doi:10.1007/s00010-007-2885-7

Alexandrov-Zeeman type theorems expressed in terms of definability

Victor Pambuccian

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

We formulate two remarkably elementary variants of theorems of the Alexandrov-Zeeman type, stated as definability statements, in minimalist axiomatic settings. The first is obtained by providing positive definitions for the notions of collinearity and segment congruence in terms of the notion of connectedness by a light ray, and is the logical counterpart of results stated in terms of mappings in [24] and [9]. The second is obtained by noticing that, in essence, results stating that mappings which preserve time-like lines preserve all lines, are the model-theoretic counterparts of definability results regarding the partial affine spaces introduced in [18].

Original language	English (US)
Pages (from-to)	249-261
Number of pages	13
Journal	Aequationes Mathematicae
Volume	74
Issue number	3
DOIs	https://doi.org/10.1007/s00010-007-2885-7
State	Published - Dec 2007
Externally published	Yes

Keywords

Alexandrov-Zeeman theorem
Definability
Minkowski space

ASJC Scopus subject areas

General Mathematics
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1007/s00010-007-2885-7

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title = "Alexandrov-Zeeman type theorems expressed in terms of definability",

abstract = "We formulate two remarkably elementary variants of theorems of the Alexandrov-Zeeman type, stated as definability statements, in minimalist axiomatic settings. The first is obtained by providing positive definitions for the notions of collinearity and segment congruence in terms of the notion of connectedness by a light ray, and is the logical counterpart of results stated in terms of mappings in [24] and [9]. The second is obtained by noticing that, in essence, results stating that mappings which preserve time-like lines preserve all lines, are the model-theoretic counterparts of definability results regarding the partial affine spaces introduced in [18].",

keywords = "Alexandrov-Zeeman theorem, Definability, Minkowski space",

author = "Victor Pambuccian",

note = "Funding Information: Acknowledgement. The results in this paper were obtained in June 2005, while the author was a guest of Prof. Dr. Karl Strambach at the Institute of Mathematics of the University of Erlangen. I thank Professor Strambach for the hospitality. Professor Istv{\'a}n N{\'e}meti{\textquoteright}s invitation to attend the Logic in Hungary conference in August 2005 provided the impetus to finalize this research, which was in part supported by an SRCA grant from ASU West. I owe to Prof. Dr. Armin Herzer a simplification of the definition of λ for Σ3.",

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doi = "10.1007/s00010-007-2885-7",

language = "English (US)",

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journal = "Aequationes Mathematicae",

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AU - Pambuccian, Victor

N1 - Funding Information: Acknowledgement. The results in this paper were obtained in June 2005, while the author was a guest of Prof. Dr. Karl Strambach at the Institute of Mathematics of the University of Erlangen. I thank Professor Strambach for the hospitality. Professor István Németi’s invitation to attend the Logic in Hungary conference in August 2005 provided the impetus to finalize this research, which was in part supported by an SRCA grant from ASU West. I owe to Prof. Dr. Armin Herzer a simplification of the definition of λ for Σ3.

PY - 2007/12

Y1 - 2007/12

N2 - We formulate two remarkably elementary variants of theorems of the Alexandrov-Zeeman type, stated as definability statements, in minimalist axiomatic settings. The first is obtained by providing positive definitions for the notions of collinearity and segment congruence in terms of the notion of connectedness by a light ray, and is the logical counterpart of results stated in terms of mappings in [24] and [9]. The second is obtained by noticing that, in essence, results stating that mappings which preserve time-like lines preserve all lines, are the model-theoretic counterparts of definability results regarding the partial affine spaces introduced in [18].

AB - We formulate two remarkably elementary variants of theorems of the Alexandrov-Zeeman type, stated as definability statements, in minimalist axiomatic settings. The first is obtained by providing positive definitions for the notions of collinearity and segment congruence in terms of the notion of connectedness by a light ray, and is the logical counterpart of results stated in terms of mappings in [24] and [9]. The second is obtained by noticing that, in essence, results stating that mappings which preserve time-like lines preserve all lines, are the model-theoretic counterparts of definability results regarding the partial affine spaces introduced in [18].

KW - Alexandrov-Zeeman theorem

KW - Definability

KW - Minkowski space

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JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

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ER -

Alexandrov-Zeeman type theorems expressed in terms of definability

Abstract

Keywords

ASJC Scopus subject areas

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