arXiv subjects
Publications and source records attributed to Alf Onshuus.
In the first part of the paper we study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. Then we try to develop analogous theory for arbitrary dependent theories.
We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite rank. We prove that such theories are coordinatised by thorn-minimal types and that such a type is unstable if an only if every non-algebraic extension thereof is. We conclude that a type is stable if and only if it admits a coordinatisation in thorn-minimal stable types. We also show that non-trivial thorn-minimal stable types extend stable sets.
Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.
We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The main tool is a theory for intersection of curves which we develop.
In a previous paper we developed the notions of th-independence and þ-ranks which define a geometric independence relation in a class of theories which we called ``rosy''. We proved that rosy theories include simple and o-minimal theories and that for any theory for which the stable forking conjecture was true, þ-forking coincides with forking independence. In this article, we continue to study properties of th-forking and find more examples of rosy theories. Among the new properties we prove in this paper are some alternative characterizations of rosy theories and some tools to prove and analyze rosiness in particular cases. Finally, we use this to find two examples of rosy non simple theories: pseudo real closed fields (PRC-fields) and the uniform companion of a large differential field defined by Marcus Tressl.
We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric structure in these models. We also prove that this definition agrees with the usual independence notions in stable, supersimple and o-minimal theories. Finally, many of the proofs and results we get for th-independence when restricted to simple theories seem to show there is some connection between th-independence and the stable forking conjecture. In particular, we prove that in any simple theory where this conjecture holds, our definition agrees with the classic definition.