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Abstract theory of semiorderings

Published online by Cambridge University Press:  17 April 2009

Thomas C. Craven  and
Tara L. Smith
Thomas C. Craven
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822, United States of America e-mail: tom@math.hawaii.edu
Tara L. Smith
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221–0025, United States of America e-mail: tsmith@math.uc.edu
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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 2 , October 2005 , pp. 225 - 250
Copyright
Copyright © Australian Mathematical Society 2005

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