Abstract
Let ( D, ∗) be a ∗-field with [ D: Z( D)] being finite. Our main objective is to show that the space of all Baer orderings (resp. weak ∗-orderings) of ( D, ∗) satisfies the strong approximation property iff every Baer ordering of ( D, ∗) is in fact a weak ∗-ordering. This shows that the notions of Baer orderings and weak ∗-orderings are respectively the "correct" analogues for semiorderings and orderings. We also intro-duce the concept of Baer formally real ∗-fields and Baer preorderings. We prove that a ∗-field admits a Baer ordering iff it is Baer formally real. In addition, some new results on weak ∗-orderings are also discussed.
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