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Almost free algebras

The essentially non-free spectrum is the class of uncountable cardinals kappa in which there is an essentially non-free algebra of cardinality kappa which is almost free. In L, the essentially non-free spectrum of a variety is entirely determined by whether or not the construction principle holds. In ZFC may be more complicated. For some varieties, such as groups, abelian groups or any variety of modules over a non-left perfect ring, the essentially non-free spectrum contains not only aleph_1 but aleph_n for all n>0. The reason for this being true in ZFC (rather than under some special set theoretic hypotheses) is that these varieties satisfy stronger versions of the construction principle. We conjecture that the hierarchy of construction principles is strict, i.e., that for each n>0 there is a variety which satisfies the n-construction principle but not the n+1-construction principle. In this paper we will show that the 1-construction principle does not imply the 2-construction principle. We prove that, assuming the consistency of some large cardinal hypothesis, it is consistent that a variety has an essentially non-free almost free algebra of cardinality aleph_n if and only if it