Towards a finer classification of strongly minimal sets.

Let M be strongly minimal and constructed by a 'Hrushovski construction' with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class T (μ triples) we show that for independent I with | I | > 1 , dcl ⁎ (I) = ∅ (* means not in dcl of a proper subset). This implies the only definable truly n -ary functions f (f 'depends' on each argument), occur when n = 1. We prove for Hrushovski's original construction and for the strongly minimal k -Steiner systems of Baldwin and Paolini that the symmetric definable closure, sdcl ⁎ (I) = ∅ (Definition 2.7). Thus, no such theory admits elimination of imaginaries. As, we show that in an arbitrary strongly minimal theory, elimination of imaginaries implies sdcl ⁎ (I) ≠ ∅. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if k = p n. The case structure depends on properties of the Hrushovski μ -function. The proofs depend on our introduction, for appropriate G ⊆ aut (M) (setwise or pointwise stabilizers of finite independent sets), the notion of a G -normal substructure A of M and of a G -decomposition of any finite such A. These results lead to a finer classification of strongly minimal structures with flat geometry, according to what sorts of definable functions they admit. [ABSTRACT FROM AUTHOR]