By Alexander Ziwet, Louis Allen Hopkins

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Is total. 2 that a field if E 2 -aI =I (-) a a > 0. E • admits an order f C E, if and only We claim for any such order vp = Dp. is formally real. 0 < a < b, then is a maximal order, 0 < £D a< -, since Thus if pa = q, It follows from $ = Dp Namely, if = (b-a)/ab = ab(b-a)/(ab) 2 > 0, and (a - Dh p,q > 0, then 1 a = (-)q > 0. p that any partial order ^ on a field E is the intersection of the total orders which refine *p. Thus any element a E E with a £ *p and some refinement. -a ^ *p can be made either positive or negative in In particular, this applies to the weak order p = 2 {2 a |a.

F(x) < 0 x E X. for at least one point X •> IR be a nonnegative continuous function with p: p E 0 f ^ '£, that is actually has only this one order outside some small neighborhood of -f G Dp . x. This proves that the order ty on Then C(X) p(x) > 0 + pf G (-£ ) and and admits no simple refinements. (4) Let functions on M M. be a C°°-manifold, The affine order is C°°(M) the ring of C°° real ^ M = {f G C°°(M) |f (X) :> 0 all x G M}. Just as in the case above, *&.. admits no refinements since there always M exist C -functions on M positive at a point and vanishing outside any small neighborhood of the point.

I) (A/P) (ii) is a semi-field. Every homomorphism of (iii) (A,*P) in (POR) is zero or injective. (0) C A are the only convex ideals, (iv) For all a G A, a ± 0, we have 1 G H(a). (v) For all a G A, a ± 0, we have 1 £ pa 2 , some Proof. The implications are trivial. 4 gives In general, if Q C A (iv) => (v). 2 has the following corollary. 3. Then • is a convex ideal, Q / A, then convex ideal if and only if every homomorphism of zero or injective. pef. a G A, a convex ideal, Q i A. (A/Q, ^/Q) a £ Q, there should exist b G A is a semi-field.