Stable generic structures

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Abstract

Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable ℵ0-categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwin's almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In particular, the generic structure need not be ω-saturated and so the argument for stability is significantly more complicated. We further show that these structures are “flat” and do not interpret a group.

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1

Partially supported by NSF grant 90000139; these results were the basis of a lecture series in Japan and the support of Tsukuba University is gratefully acknowledged.

2

Portions of this paper are from Shi's 1992 Ph.D. Thesis at the University of Illinois in Chicago.