On inverse gamma-systems and the number of L-infinity lambda-equivalent, non-isomorphic models for lambda singular

Suppose lambda is a singular cardinal of uncountable cofinality kappa. Tor a model M of cardinality lambda. let No (M) denote the number of isomorphis m types of models N of cardinality lambda which are L-infinity lambda-equiv alent to M. In [7] Shelah considered inverse kappa-systems A of abelian gro ups and their certain kind of quotient limits Gr(A)/ Fact(A). In particular Shelah proved in [7. Fact 3.10] that for every cardinal mu there exists an inverse kappa-system A such that A consists of abelian groups having cardi nality at most mu(kappa) and card (Gr(A)/Fact(A)) = mu. Later in [8, Theore m 3.3] Shelah showed a strict connection between inverse K-systems and poss ible values of No (under the assumption that 0(kappa) < lambda for every 0 < lambda): if A is an inverse kappa-system of abelian groups having cardina lity < lambda, then there is a model M such that card (M) = lambda and No(M ) = card(Gr(A)/Fact(A)). The following was an immediate consequence (when 0 (kappa) < lambda for every 0 < lambda): for every nonzero mu < lambda or mu = lambda(kappa) there is a model M-mu of cardinality lambda with No(M-mu) = mu. In this paper we show: for every nonzero mu less than or equal to lam bda(kappa) there is an inverse kappa-system A of abelian groups having card inality < lambda such that card(Gr(A)/Fact(A)) = mu(under the assumptions 2 (kappa) < lambda and 0 (<)kappa <lambda for all 0 < lambda when mu > lambda ), with the obvious new consequence concerning the possible value of No. Sp ecifically. the case No(M) = lambda is possible when 0(kappa) < lambda for every 0 < lambda.