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dc.contributor.advisorMarshall, Murrayen_US
dc.creatorGladki, Pawelen_US
dc.date.accessioned2007-09-17T13:52:19Zen_US
dc.date.accessioned2013-01-04T04:58:51Z
dc.date.available2007-09-18T08:00:00Zen_US
dc.date.available2013-01-04T04:58:51Z
dc.date.created2007-09en_US
dc.date.issued2007-09-18en_US
dc.date.submittedSeptember 2007en_US
dc.identifier.urihttp://hdl.handle.net/10388/etd-09172007-135219en_US
dc.description.abstractThe 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.isoen_USen_US
dc.subjectspaces of orderingsen_US
dc.subjectforms over real fieldsen_US
dc.titleThe pp conjecture in the theory of spaces of orderingsen_US
thesis.degree.departmentMathematics and Statisticsen_US
thesis.degree.disciplineMathematics and Statisticsen_US
thesis.degree.grantorUniversity of Saskatchewanen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophy (Ph.D.)en_US
dc.type.materialtexten_US
dc.type.genreThesisen_US


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