Note on the question of Sikora

A natural topology on the set of left orderings on free abelian groups and free groups [Formula: see text], [Formula: see text] has studied in [A. S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36(4) (2004) 519–526; L. Smith, On ordering free groups, J. Symbolic Comput. 40 (2005) 1285–1290, Corrigendum (with A. Clay) 44 (2009) 1529–1532]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a conjecture: this is also true for the free group [Formula: see text] with [Formula: see text] generators. We point out the paper dealing with equivalent questions.

A note on the pp conjecture for sheaves of spaces of orderings

In this note we provide a direct and simple proof of a result previously obtained by Astier stating that the class of spaces of orderings for which the pp conjecture holds true is closed under sheaves over Boolean spaces.

THE pp CONJECTURE FOR SPACES OF ORDERINGS OF RATIONAL CONICS

First counterexamples are given to a basic question raised in [10]. The paper considers the space of orderings (X,G) of the function field of a real irreducible conic [Formula: see text] over the field ℚ of rational numbers. It is shown that the pp conjecture fails to hold for such a space of orderings when [Formula: see text] has no rational points. In this case, it is shown that the pp conjecture "almost holds" in the sense that, if a pp formula holds on each finite subspace of (X,G), then it holds on each proper subspace of (X,G). For pp formulas which are product-free and 1-related, the pp conjecture is known to be true, at least if the stability index is finite [11]. The counterexamples constructed here are the simplest sort of pp formulas which are not product-free and 1-related.

Abstract theory of semiorderings

Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.