Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 2 10.wxyz/CV72P2 http://www.austms.org.au/Publ/Bulletin/V72P2/ Thomas C. Craven Tara L. Smith 14 February 2006 2006 2 14 225 250 722-5102-CrSm-2005 10.wxyz/C2005V72P2p225 http://www.austms.org.au/Publ/Bulletin/V72P2/722-5102-CrSm/ 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. 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