Boris Zil'ber conjectured that all strongly minimal theories are bi-interpretable with one of the “classical” sorts: theories of algebraically closed fields, theories of infinite vector spaces over division rings and theories with trivial algebraic closure relations. Hrushovski produced the first two classes of counterexample to this conjecture in [10] and [9]. Subsequently, in [8], the author gave an explicit axiomatization of a special case of [9] from which model completeness could quickly be deduced. It was unclear at that writing whether the model completeness result was true in the general case or was due to peculiarities of the case under consideration. The main new result of this paper is model completeness, not only of the general case in [9], but also of the theories described in [10]. Specifically, we present a general framework in which producing a strongly minimal theory is reduced to finding an elementary class of theories satisfying certain requirements (see below). We present the theories of [10] and [9] as special instances of such theories, giving an explicit axiomatization from which model completeness immediately follows in each case.

We hope by presenting these constructions in parallel, using common language and extracting common elements, to make easier both the exploitation of the ideas involved in their making and their comparison with other recent constructions of a similar flavor. For a selection of such constructions, see [6], [1], [2] and [3]. For more general background, see [2], [4] and [11].