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) < 2λ for some λ > ∣∣. If I(K,λ) = 2λ 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.Before we give the definition of strongly abelian, let us motivate how it arises in this context. A variety is locally finite if every finitely generated algebra in is finite. If and are subvarieties of then = ⊗ means that is the variety generated by and and moreover there is a term τ(x, y) so that τ(x, y) = x holds in and τ(x, y) = y holds in . is called the varietal product of and . As a consequence, if = ⊗ then for every M ∈ there is a unique (up to isomorphism) A ∈ and B ∈ so that M ≅ A × B.In [4], McKenzie and Valeriote proved the following theorem.Theorem 0.1. Ifis a locally finite decidable variety, then there are three subvarieties of, , and, so that = ⊗ ⊗ andis an affine variety, is a strongly abelian variety andis a discriminator variety.For the exact definitions of the terms affine and discriminator one can see [4]; however, for us here it is important to know that an affine variety is polynomially equivalent to a variety of left R-modules over some ring R and that any nontrivial discriminator variety contains an algebra whose complete theory is unstable.