The notion of an almost strongly minimal theory was introduced in [1]. Such a theory is a particularly simple sort of ℵ1-categorical theory. In [1] we characterized this simplicity in terms of the Stone space of models of T. Here, we characterize almost strongly minimal theories which are not ℵ0-categorical in terms of D. M. R. Park's notion [4] of a theory with the strong elementary intersection property. In addition we prove a useful sufficient condition for an elementary theory to be an almost strongly minimal theory. Our notation is from [1] but this paper is independent of the results proved there. We do assume familiarity with §1 and §2 of [2].

In [4], Park defines a theory T to have the strong elementary intersection property (s.e.i.p.) if for each model C of T and each pair of elementary submodels of C either is an elementary submodel of C. T has the nontrivial strong elementary intersection property (n.s.e.i.p.) if for each triple C, as above Park proves the following two statements equivalent: