A structure theorem for strongly abelian varieties with few models

By a variety, we mean a class of structures in some language containing only function symbols which is equationally defined or equivalently is closed under homomorphisms, submodels and products. If K is a class of -structures then I ( K , λ) denotes the number of nonisomorphic models in K of cardinality λ. When we say that K has few models, we mean that I ( K ,λ) < 2 λ for some λ > ∣ ∣. If I ( K ,λ) = 2 λ for all λ > ∣ ∣, then we say K has many models. In [9] and [10], Shelah has shown that for an elementary class K , having few models is a strong structural condition.