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 mu is in a certain class T (mu triples) we show that for independent I with |I| > 1, dcl*(I) = 0 (* 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) = & empty; (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) not equal & empty;. 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 mu-function. The proofs depend on our introduction, for appropriate G subset of 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. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This article studies the strongly minimal M constructed by a 'Hrushovski construction', and shows that if the Hrushovski algebraization function μ belongs to a certain class T, then for independent I, dcl*(I) = 0. This implies that the only definable truly n-ary functions f occur when n = 1. It is also demonstrated that for Hrushovski's original construction and the strongly minimal k-Steiner systems of Baldwin and Paolini, the symmetric definable closure sdcl*(I) = ∅, indicating the impossibility of eliminating imaginaries.