Weakly *-ordered *-fields Original Research Article

Let (D, *) be a weakly *-ordered *-field and F′ the field of symmetric elements in the center of D. We find a necessary and sufficient condition to extend an ordering of F′ to a weak *-ordering of (D, *). Using this, we prove that our notion of weak preorderings is the correct generalization of preorderings in ordered fields. We then study the notion of forms over weak preorderings and SAP preorderings. As a result, we can also determine when a semiordering of F′ be extendable to a Baer ordering in (D, *).