The notion of an almost strongly minimal theory was introduced in . Such a theory is a particularly simple sort of ℵ1-categorical theory. In  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  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  but this paper is independent of the results proved there. We do assume familiarity with §1 and §2 of .
In , 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: