Multigroups and the foundations of geometry
type: Item
instance of: scholarly article
title: Multigroups and the foundations of geometry
zbMATH Open document ID: 0731.51009
DOI: 10.1007/BF02846360
author: A. Barlotti
author: Karl Strambach
publication date: 1991-01-01T00:00:00Z
review text: Multigroups are sets with a binary commutative and associative operation with values in the power set, cf. \textit{W. Prenovitz} and \textit{J. Jantosciak} [Join Geometries. A theory of convex sets and linear geometry. New York etc.: Springer-Verlag (1979; Zbl 0421.52001)]. The geometry \(G(M)\) associated to a multigroup \(M\) (with identity e) has point set \(M\setminus \{e\}\), and the lines are submultigroups generated by two independent elements of \(M\setminus \{e\}.\) The authors consider mainly three types of (geometric) multigroups. The descriptive multigroups axiomatize the convex (open) regions \(C\) in ordered projective spaces with the property that every line meeting \(C\) in an open set has non-empty intersection with the boundary of \(C\). The multigroups associated with (arbitrary) projective spaces are called projective multigroups. The spherical multigroups are associated with spherical geometries, which can be obtained from ordered projective spaces by a doubling procedure, as the authors show (3.1, 3.2), thus refining results of \textit{W. Prenowitz} [Can. J. Math. 2, 100--119 (1950; Zbl 0041.27203)]. This fact allows to translate the theory of ordered projective spaces into the language of multigroups. Indeed, the authors show that the foundations of geometry have important applications to (geometric) multigroups. Many of those applications depend on the observation (1.2, 1.3, 1.4) that the automorphisms of a geometric multigroup \(M\) are precisely the collineations of the geometry \(G(M)\). Section 4 of the paper deals mainly with descriptive multigroups \(S\) and their geometries, which are assumed to be separable and connected (in the order topology). If \(S\) is finitely generated, then \(S\) is homeomorphic to some \(\mathbb R^ n\), and the group \(\Gamma\) of all automorphisms of \(S\) is a Lie group (4.3). Suitable transitive assumptions on \(\Gamma\) lead to characterizations of real affine spaces and of real hyperbolic spaces (4.5, 4.…
reviewed by: Theo Grundhöfer
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 20N99
Mathematics Subject Classification ID: 51D10
Mathematics Subject Classification ID: 51A30
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 4208958
zbMATH Keywords: ordered projective spaces
zbMATH Keywords: multigroups
MaRDI profile type: Publication
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (cites work)
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Multigroups and the foundations of geometry (published in)
published in: Rendiconti del Circolo Matematico di Palermo