In 1972 A.H. Lachlan asked whether all stable ℵ0-categorical theories were also ω-stable. This claim was refuted by E. Hrushovski when he constructed a counter-example to the conjecture. It is a generalisation of this counterexample which provides the structures forming the basis of this thesis. We refer to these structures as Hrushovski constructions. We begin by looking at how such constructions fit in to Shelah’s strong order properties. D. Evans proved that under certain assumptions the theories of Hrushovski’s constructions are simple. He then suggested that by dropping a condition ensuring simplicity, these constructions could be composed in a way that its theories failed both simplicity and strong order property 3. As we will show however, it turns out that this condition actually provides a dividing line between simplicity and strong order property 3. The next section considers the ‘oak’ property on Hrushovski constructions. As oak and strong order property 3 both imply non-simplicity in this class, we investigate the effect that the dividing line between simplicity and strong order property 3 has on oak in Hrushovski constructions. Our investigations provide us with a version of independence specific to Hrushovski constructions. We finish off by showing how this relates to standard notions of independence such as Shelah’s forking-independence and Onshuus’s þ-independence.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:439839 |
| Date | January 2007 |
| Creators | Wong, Mark Wing Ho |
| Publisher | University of East Anglia |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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