dc.contributor.advisor Marshall, Murray en_US dc.creator Gladki, Pawel en_US dc.date.accessioned 2007-09-17T13:52:19Z en_US dc.date.accessioned 2013-01-04T04:58:51Z dc.date.available 2007-09-18T08:00:00Z en_US dc.date.available 2013-01-04T04:58:51Z dc.date.created 2007-09 en_US dc.date.issued 2007-09-18 en_US dc.date.submitted September 2007 en_US dc.identifier.uri http://hdl.handle.net/10388/etd-09172007-135219 en_US dc.description.abstract The notion of spaces of orderings was introduced by Murray Marshall in the 1970's and provides an abstract framework for studying orderings on fields and the reduced theory of quadratic forms over fields. The structure of a space of orderings (X, G) is completely determined by the group structure of G and the quaternary relation (a_1, a_2) = (a_3, a_4) on G -- the groups with additional structure arising in this way are called reduced special groups. The theory of reduced special groups, in turn, can be conveniently axiomatized in the first order language L_SG. Numerous important notions in this theory, such as isometry, isotropy, or being an element of a value set of a form, make an extensive use of, so called, positive primitive formulae in the language L_SG. Therefore, the following question, which can be viewed as a type of very general and highly abstract local-global principle, is of great importance:Is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space?This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this thesis, we discuss, discovered by us, first counterexamples for which the pp conjecture fails. Namely, we classify spaces of orderings of function fields of rational conics with respect to the pp conjecture, and show for which of such spaces the conjecture fails, and then we disprove the pp conjecture for the space of orderings of the field R(x,y). Some other examples, which can be easily obtained from the developed theory, are also given. In addition, we provide a refinement of the result previously obtained by Vincent Astier and Markus Tressl, which shows that a pp formula fails on a finite subspace of a space of orderings, if and only if a certain family of formulae is verified. en_US dc.language.iso en_US en_US dc.subject spaces of orderings en_US dc.subject forms over real fields en_US dc.title The pp conjecture in the theory of spaces of orderings en_US thesis.degree.department Mathematics and Statistics en_US thesis.degree.discipline Mathematics and Statistics en_US thesis.degree.grantor University of Saskatchewan en_US thesis.degree.level Doctoral en_US thesis.degree.name Doctor of Philosophy (Ph.D.) en_US dc.type.material text en_US dc.type.genre Thesis en_US
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