Valuation theory of higher level <f>*</f>-signatures

Around 1980 two generalizations of the theory of linearly ordered fields appeared in the literature: Becker''s theory of orderings of higher level on fields (J. Reine Angew. Math. 307/308 (1979)8) and Holland''s theory of <f>*</f>-orderings on skew-fields with involution (J. Algebra 101 (1) (1986) 16–46). The aim of this paper is to unify both theories.In Section 1 we define (higher level) <f>*</f>-signatures on domains with involution which correspond to higher level preorderings in Becker''s theory. The subclasses of 2-cyclic and cyclic <f>*</f>-signatures correspond to complete preorderings and orderings respectively. We prove a necessary and sufficient condition for extendability of <f>*</f>-signatures from Ore domains to skew-fields of fractions.In Section 2 we define the set of bounded elements of a <f>*</f>-signature on a skew field with involution. If the skew field contains a central element <f>i</f> such that <f>i2=-1</f> and <f>i*=-i</f> and the <f>*</f>-signature is 2-cyclic then the set of bounded elements is an invariant valuation ring. An example shows that the assumption on <f>i</f> cannot be omitted.In Section 3 we define extended <f>*</f>-signatures and prove that every 2-cyclic <f>*</f>-signature on a skew field <f>D</f> with <f>i∈Z(D)</f> is a restriction of some extended <f>*</f>-signature.In Section 4 we define extended <f>*</f>-preorderings as positive cones of extended <f>*</f>-signatures. We show that every <f>*</f>-preordering which is a restriction of an extended <f>*</f>-preordering is equal to the intersection of all <f>*</f>-orderings containing it. The assumption <f>i∈Z(D)</f> is not required.Section 5 presents auxilliary material for the proof of the weak isotropy principle for higher level <f>*</f>-signatures which is given in Section 6. [Copyright &y& Elsevier]