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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: [email protected]
Tara L. Smith
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221–0025, United States of America e-mail: [email protected]
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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

References

[1]Andradas, C., Bröcker, L. and Ruiz, J., Constructible sets in real geometry (Springer Verlag, Berlin, 1966).Google Scholar
[2]Baer, R., Linear algebra and projective geometry (Academic Press, New York, 1952).Google Scholar
[3]Becker, E. and Köpping, E., ‘Reduzierte quadratische Formen und semiornungen reeller Körpern’, Abh. Math. Sem. Univ. Hamburg 46 (1977), 143–177.CrossRefGoogle Scholar
[4]Bröcker, L., ‘Zur Theorie der quadratischen Formen über formal reellen Körpern’, Math. Ann. 210 (1974), 233–256.CrossRefGoogle Scholar
[5]Cohn, P.M., Skew fields, Encyclopedia of Mathematics and its Applications 57 (Cambridge Univ. Presspubladdr Cambridge, 1995).CrossRefGoogle Scholar
[6]Craven, T., ‘The Boolean space of orderings of a field’, Trans. Amer. Math. Soc. 209 (1975), 225–235.CrossRefGoogle Scholar
[7]Craven, T., ‘Characterizing reduced Witt rings of fields’, J. Algebra 53 (1978), 68–77.CrossRefGoogle Scholar
[8]Craven, T., ‘An application of Pólya's theory of counting to an enumeration problem arising in quadratic form theory’, J. Combin. Theory Ser. A 29 (1980), 174–181.Google Scholar
[9]Craven, T., ‘Orderings and valuations on ✶-fields’, Rocky Mountain J. Math. 19 (1989), 629–646.CrossRefGoogle Scholar
[10]Craven, T., ‘Witt groups of hermitian forms over ✶-fields’, J. Algebra 147 (1992), 96–127.CrossRefGoogle Scholar
[11]Craven, T., ‘*-valuations and hermitian forms on skew fields’, in Valuation Theory and its Applications 1, Fields Inst. Commun. (American Mathematical Society, Providence, RI, 2002).Google Scholar
[12]Craven, T. and Smith, T., ‘Formally real fields from a Galois theoretic perspective’, J. Pure Appl. Alg. 145 (2000), 19–36.CrossRefGoogle Scholar
[13]Craven, T. and Smith, T., ‘Hermitian forms over ordered *-fields’, J. Algebra 216 (1999), 86–104.CrossRefGoogle Scholar
[14]Craven, T. and Smith, T., ‘Semiorderings and Witt rings’, Bull. Austral. Math. Soc. 67 (2003), 329–341.Google Scholar
[15]Efrat, I. and Haran, D., ‘On Galois groups over pythagorean semi-real closed fields’, Israel J. Math. 85 (1994), 57–78.CrossRefGoogle Scholar
[16]Jacobi, T. and Prestel, A., ‘Distinguished representations of strictly positive polynomials’, J. Reine Angew. Math. 532 (2001), 223–235.Google Scholar
[17]Kleinstein, J. and Rosenberg, A., ‘Signatures and semisignatures of abstract Witt rings and Witt rings of semilocal rings’, Canad. J. Math. 30 (1978), 872–895.Google Scholar
[18]Kleinstein, J. and Rosenberg, A., ‘Succinct and representational Witt rings’, Pacific J. Math. 86 (1980), 99–137.CrossRefGoogle Scholar
[19]Knebusch, M., ‘Generalization of a theorem of Artin-Pfister to arbitrary semilocal rings, and related topics’, J. Algebra 36 (1975), 46–67.Google Scholar
[20]Knebusch, M., Rosenberg, A. and Ware, R., ‘Structure of Witt rings and quotients of abelian group rings’, Amer. J. Math. 94 (1972), 119–155.CrossRefGoogle Scholar
[21]Lam, T. Y., Orderings, valuations and quadratic forms, Conference Board of the Mathematical Sciences 52 (American Mathemtical Society, Providence, RI, 1983).CrossRefGoogle Scholar
[22]Marshall, M., Abstract Witt rings Queen's Papers in Pure and Appl. Math. 57 (Queen's University, Kingston, Ontario, 1980).Google Scholar
[23]Marshall, M., Spaces of orderings and abstract real spectra, Lecture Notes in Math. 1636 (Springer-Verlag, Berlin, 1996).Google Scholar
[24]Merzel, J., ‘Quadratic forms over fields with finitely many orderings’, Contemp. Math. 8 (1982), 185–229.CrossRefGoogle Scholar
[25]Prestel, A., Lectures on formally real fields, Lecture Notes in Math. 1093 (Springer-Verlag, Berlin, 1984).CrossRefGoogle Scholar
[26]Prestel, A. and Delzell, C., Positive polynomials: from Hilbert's 17th problem to real algebra, London Math. Soc. Note Ser. 36 (Springer-Verlag, Berlin, 2001).Google Scholar