Universität zu Köln

Struve, Rolf and Struve, Horst (2019). The Thomsen-Bachmann correspondence in metric geometry I. J. Geom., 110 (1). BASEL: SPRINGER BASEL AG. ISSN 1420-8997

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Abstract

We study in this two part paper the Thomsen-Bachmann correspondence between metric geometries and groups which is often summarized by the phrase 'Geometry can be formulated in the group of motions'. We show that (1) the correspondence can be precisely stated in a framework of first-order logic, (2) the correspondence, which was established by Thomsen and Bachmann for Euclidean and for plane absolute geometry, holds also for Hjelmslev geometries, Cayley-Klein geometries, isotropic and equiform geometries, and (3) these geometries and the theory of their group of motions are not only mutually interpretable but also bi-interpretable. Hence a reflection-geometric axiomatization of a class of motion groups corresponds to an elementary axiomatization of the underlying geometry and provides with the calculus of reflections a powerful proof method. In the first part of the paper we introduce the fundamental logical and geometric notions and show that the Thomsen-Bachmann correspondence can be rephrased in first-order logic by 'The theory of the group of motions is a conservative extension of the underlying geometric theory'.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Struve, Rolf UNSPECIFIED UNSPECIFIED UNSPECIFIED Struve, Horst UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-152807
DOI:	10.1007/s00022-018-0465-8
Journal or Publication Title:	J. Geom.
Volume:	110
Number:	1
Date:	2019
Publisher:	SPRINGER BASEL AG
Place of Publication:	BASEL
ISSN:	1420-8997
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/15280