Cyclic order: a geometric analysis
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Cyclic order: a geometric analysis Rolf Struve1 Received: 9 January 2020 / Accepted: 13 February 2020 © The Managing Editors 2020
Abstract Concepts of order play an important role in many branches of mathematics. We start the article with an analysis of the notion of cyclic order in algebraic structures, which includes a characterization of cyclically ordered groups by cyclic cones and the introduction of the notion of a cyclically ordered field. We then study the role of cyclic order in the foundations of geometry. In Euclidean and in absolute geometry, order structures are introduced by linear orders (see Hilbert in Grundlagen der Geometrie, Teubner, Stuttgart, 1972; Coxeter in Introduction to geometry, Wiley, New York, 1961; Sperner in Beziehungen zwischen geometrischer und algebraischer Anordnung. Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 1949; Bachmann in Aufbau der Geometrie aus dem Spiegelungsbegriff, Springer, Heidelberg, 1973; H Struve and R Struve in J Geom 105:419–447, 2014; R Struve in J Geom 106:551–570, 2015). This excludes elliptic geometry. We show that the notion of cyclic order (on pencils of lines) allows the introduction of order structures in a unified way (including the elliptic case) and corresponds on the algebraic side to a linear order of the associated coordinate field. In addition we prove that the three classical geometries (Euclidean, hyperbolic, and elliptic) over fields K of characteristic = 2 are orderable if and only if a separation relation on rows of collinear points and a separation relation on pencils of concurrent lines can be defined which are ‘compatible’. The article closes with a geometric interpretation of cyclically ordered fields as Gaußian coordinate fields of Euclidean Hilbert planes. Keywords Cyclic order · Cyclically ordered group · Cyclically ordered field · Cyclic cone · Ordered geometry · Absolute geometry · Ordered Bachmann groups Mathematics Subject Classification 51G05 · 51F05 · 51F15 · 12J15
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Rolf Struve [email protected] Auf der Panne 36, 44805 Bochum, Germany
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Beitr Algebra Geom
1 Introduction “A discussion of order ... has become essential to any understanding of the foundations of mathematics” observed Russell in The principles of Mathematics (Russell 1903, p. 199), since they allow an explication of infinity, continuity and other fundamental concepts of Arithmetic, Analysis and Geometry. According to Russell (1903, p. 200) there are two types of order: a linear order which can be described by a ternary betweenness relation, and a cyclic order, which can be described by a quaternary separation relation. A betweenness relation corresponds to a pair () of dual binary relations (which are asymmetric, transitive and total) and a separation relation corresponds to a pair of dual ternary relations (, ∗ ), which are cyclic, asymmetric, transitive and total.1 From a geometric point of view, linear and cyclic order are fundamental ideas of our intuition of space (in Euclidean geometry the points on a line are linearly ordere
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