Ordered Groups by Joseph Muscat

By Joseph Muscat

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3. R[x], R[[x]], R((x)) with lex ordering. The subring of terms M n=−N an xn . 4. g. Z to Q. Properties: 1. Equivalently, they are the indecomposable lattice rings (no proper idempotents). (1 − e) so e = e2 Proof: For any idempotent, either e so 1 − e 0. 2. ax ay ⇒ x 0 ⇒ x y if a > 0, else a ax = 0 (a = 0) ⇒ |x| < 1 (else |x| 0 or (1 − e) e y. 1 ⇒ |a| |a||x| = |ax| = 0). 3. Recall that linear orders have a natural T5 topology; which is connected iff order-complete and without cuts or gaps. 4. Reduced linearly ordered rings are domains.

Henriksen, “A survey of f-rings and some of their generalizations” 2.

2. x → ax for a > 0 are precisely the (+, )-automorphisms. The only (+, ∗, )-automorphism is trivial. 3. If x y + a for all a > 0, then x y (else x − y a := (x − y)/2). Joseph Muscat 2015 31 4. A field can be linearly ordered ⇔ it can be lattice-ordered ⇔ it is formally real. Proof: A formally real field can have its positives P extended maximally to Q, by Hausdorff’s maximality principle. , −x ∈ Q. More generally a ring can be linearly ordered ⇔ proper sums of even products of elements cannot be zero (same proof).

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