Multigroups and the foundations of geometry
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, 4.6, 4.8). If 3 is the minimal number of generators of \(S\), then \(G(S)\) is a flat plane in the sense of \textit{H. R. Salzmann} [Adv. Math. 2, 1--60 (1967; Zbl 0153.21601)], and the results of flat planes of Salzmann and his school can be expressed in terms of these multigroups. The authors address the problem of classification of the multigroups \(S\) with a transitive group of automorphisms; this problem is closely related with work of \textit{È. B. Vinberg} [Sov. Math. Dokl. 2, 1470--1473 (1961); translation from Dokl. Akad. Nauk SSSR 141, 521--524 (1961; Zbl 0114.38402) and Trans. Mosc. Math. Soc. 1963, 340--403 (1965); translation from Tr. Mosk. Mat. O.-va. 12, 303--358 (1963; Zbl 0138.43301)] on convex homogeneous and \(T\)-algebras. Sections 5 and 7 introduce metric descriptive multigroups and metric projective multigroups, in the spirit of \textit{F. Bachmann} [Aufbau der Geometrie aus dem Spiegelungbegriff. 2nd ed., Springer (1973; Zbl 0254.50001)]. Section 6 is devoted to projective multigroups. First the authors study finite projective multigroups, i.e. finite projective spaces, then they turn to compact topological projective multigroups, i.e. compact topological projective space [cf. \textit{J. Misfeld}, Abh. Math. Semin. Univ. Hamb. 32, 232--263 (1968; Zbl 0164.20804)]. The compact topological projective spaces of dimension at least 3 are classified by the locally compact skew fields. The theory of compact projective planes, which was created by Salzmann, is much more complicated. Again the results of Salzmann and his school can be translated into the language of projective multigroups (with 3 as minimal number of generators).
scientific article; zbMATH DE number 4208958
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