Submission date: 29 June 2021

Abstract:

Let M be strongly minimal and constructed by a `Hrushovski construction'. 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 function f (f `depends' on each argument), occur when n=1.
We prove, indicating the dependence on μ, for Hrushovski's original construction and including analogous results for the strongly minimal k-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, sdcl^*(I) =∅, and thus the theory does not admit elimination of imaginaries. 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 proofs depend on our introduction for appropriate G ⊆ Aut(M) the notion of a G-normal substructure A of M and of a G-decomposition of A.
These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit.

Mathematics Subject Classification: 03C05 (Primary) 03C45, 03C98 (Secondary)

Keywords and phrases:

Full text arXiv 2106.15567: pdf, ps.