On inverse γ-systems and the number of L_∞λ-equivalent, non-isomorphic models for λ singular

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model Script M sign of cardinality λ. let No (Script M sign) denote the number of isomorphism types of models script N of cardinality λ which are L_∞λ-equivalent to Script M sign. In [7] Shelah considered inverse κ-systems Script A sign of abelian groups and their certain kind of quotient limits Gr(Script A sign)/ Fact(Script A sign). In particular Shelah proved in [7. Fact 3.10] that for every cardinal μ there exists an inverse κ-system Script A sign such that Script A sign consists of abelian groups having cardinality at most μ^κ and card(Gr(Script A sign)/ Fact(Script A sign)) = μ. Later in [8. Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θ^κ < λ for every θ < λ): if Script A sign is an inverse κ-system of abelian groups having cardinality < λ, then there is a model Script M sign such that card(Script M sign) = λ and No(Script M sign) = card (Gr(Script A sign)/ Fact(Script A sign)). The following was an immediate consequence (when θκ < λ for every θ < λ): for every nonzero μ < λ or μ = λ^κ there is a model Script M signμ of cardinality λ with No (Script M signμ) = μ. In this paper we show: for every nonzero μ ≤ λ^κ there is an inverse κ-system Script A sign of abelian groups having cardinality < λ such that card(Gr(Script A sign)/ Fact(Script A sign)) = μ (under the assumptions 2^κ < λ and θ^<κ < λ for all θ < λ when μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No(Script M sign) = λ is possible when θ^κ < λ for every θ < λ.