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 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 Gnormal 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.12 As, clearly sdcl(∅) = dcl(∅), which can be non-empty, and by Theorem 4.1.2 we can construct X with dcl * (X) = ∅.13 This definition is more restrictive than the standard (e.g. [Grä79, p. 35] as for our definition, in a ring the polynomial xy + z does not depend on y while usually one is allowed to substitute 0 to witness dependence.
Key wordsThe independence property, stability, dependent theories.
MSC (2010) 03C07, 03C45, 03C64A theory is stable up to Δ if any Δ-type over a model has a few extensions up to complete types. We prove that a theory has no the independence property iff it is stable up to some Δ, where each ϕ(x;ȳ) ∈ Δ has no the independence property.
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