Bulletin of the Australian Mathematical Society

Research Article

Abstract theory of semiorderings

Thomas C. Cravena1 and Tara L. Smitha2

a1 Department of Mathematics, University of Hawaii, Honolulu, HI 96822, United States of America e-mail: tom@math.hawaii.edu

a2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221–0025, United States of America e-mail: tsmith@math.uc.edu

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.

(Received March 31 2005)