β Scribed by J.R.P. Christensen; Vladimir Kanovei; Michael Reeken
No coin nor oath required. For personal study only.
We prove that any Borel Abelian ordered group B, having a countable subgroup G as the largest convex subgroup, and such that the quotient B/G is order isomorphic to R, the reals, is Borel grouporder isomorphic to the product R Γ G, ordered lexicographically. As a main ingredient of this proof, we show, answering a question of D. Marker, that all Borel cocycles R 2 β Z are Borel coboundaries. A Borel classification theorem for Borel ordered CCC groups is proved.
π SIMILAR VOLUMES
It is known that the spaces of orders on orderable computable ΓΏelds can represent all 0 1 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent 0 1 classes in even a weak manner. Next, we consider presentations of ordered
I n this paper, we introduce the concept of a formally negative set in an orderable field and we study field extensions in which certain collections of orders of the ground field may be extended to the larger field. Our methods are altogether classical and actually our definitions and results are ve