Minimal topological projective planes

Extending the classical theorem of Baer and Krull and related results to arbitrary projective planes we study the compatibility between epimorphisms and halforderings of projective planes. Further, by geometrical means, the classical constructions of finest, ordercompatible valuations and of finest, order-compatible valuations coarser than given valuations are imitated for arbitrary projective planes. This allows us to make a first attack towards trivialization of fans in projective planes.Paralleling the role order-compatible valuations play in the theory of spaces of orderings of fields, its geometrical counterpart, the order-compatible epimorphisms of planes, has turned out to be a fruitful tool in studying spaces of orderings over arbitrary projective planes. Basic work on the geometric notion of compatibility was started by Prieg-Crampe [28], and one of the central classical results, the theorem of Baer and Krull, was carried over to arbitrary planar ternary rings (PTRs) and thus to projective planes in [15]. But, in contrast to the classical situation, this result is not always applicable: there do exist non-Archimedean ordered PTRs which allow no nontrivial orderpreserving places (see Andr6 [1]) as well as non-formally-real PTRs allowing places into the reals (see Tschimmel [36]).We do two things in this paper. On the one hand, we shall go into some detail concerning the Baer-Krull theorem for projective planes. We shall establish its mechanism for Sperner's halforderings [32], [33] in Section 1, summarize characterizations of compatibility between preorderings and places and finally discuss some consequences of the Baer-Krull theorem in Section 2. On the other hand, in face of the restrictions mentioned above, we shall address the problem of the existence of order-compatible epimorphisms. By mere geometrical arguments, we show in Section 3 that the classical construction of an order-compatible epimorphism through which a given epimorphism factors extends to our setting. This allows us to study the compatibility between epimorphisms and sets of orderings in the last section, and to make a first attack towards Br6cker's trivialization of fans [4] for arbitrary projective planes. EPIMORPHISMS, PLACES AND HALFORDERINGSWe shall denote the points (lines) of a projective plane H = (~, if, I) by lower case (upper case) letters. Let GH be the intersection of two different lines G and Geometriae Dedicata 32: 59-79, 1989.

Section: On the Trivialization Of Finite Fansmentioning

confidence: 99%

Some local-global principles for ordered projective planes

Kalhoff

1989

“…Szambien [14]), das heiBt, dab es keine echt gr6bere Topologie gibt, so dab die Ebene eine topologische Ebene bleibt. Zu untersuchen bleibt die Stetigkeit an den Nahtstellen.…”

unclassified

Topologien von Moultonebenen

Hartmann

1989

Moultonebenen sind angeordnete projektive Ebenen und als solche topologische projektive Ebenen. In dieser Arbeit werden Topologien von Moultonebenen untersucht, die vonder Ordnungstopologie verschieden sein kSnnen. Solche Topologien sind stets feiner als die Ordnungstopologie der Ebene; es gibt keine kompakten unzusammenh/ingenden Moultonebenen. Im allgemeinen ist nicht bekannt, ob sich Topologien von Tern/irkSrpern stets auf die zugehSrigen projektiven Ebenen fortsetzen lassen. In Moult,onebenen ist die Topologie einer koordinatisierenden M-Gruppe so auf die Ebene fortsetzbar, dab diese eine topologische projektive Ebene wird. Moultonebenen haben viele Epimorphismen, und diese sind all bekannt. Als Anwendung des Fortsetzungssatzes wird gezeigt, dab Moultonebenen mit der durch einen Epimorphismus erzeugten Stellentopologie stets topologische Ebenen sind. Die Stellentopologie einer Moultonebene ist genau dann diskret, wenn der Epimorphismus injektiv ist.Sei K ein angeordneter Sehiefk6rper, seien d, k e K mit d ~< 0 < k ~ 1. In K wird eine neue Multiplikation o erkl/irt durchDie Struktur K(+, o) wird mit Kk,d und, falls d = 0 ist, mit K k bezeichnet. K k ist eine Cartesische Gruppe, die dariiber gebildeten Ebenen heiBen Moultonebenen.

Topological projective planes and uniform valuations

Kalhoff

1995

The valuation topology of any uniformly valued ternary field (K, T, v) can be extended to the projective plane H over (K, T) making it a topological projective plane in the sense of Salzmann. Appealing to PrieB-Crampe's celebrated fixed point theorem for ultrametric spaces, our result allows us to present a wide variety of new, totally disconnected, compact and non-compact topological projective planes. (1991). 51H 10, 51 A10, 17A80, 12K99. Mathematics Subject Classifications* Asusuallet a--}-b := T(1, a, b), ab := T(a, b, 0), K* := K\{O},andwdtea-b, -b, a/c, and c\a for the elements defined by (a -b) -t-b = a, (-b) -t-b = 0, (a/c)c = a, and c(c\a) = a respectively (a, b, c E K, c ¢ 0). ** In contrast to [8], we require a uniform valuation to be onto, excluding the trivial valuation from our considerations.