### Almost free algebras

by Mekler and Shelah. [MkSh:366]

Israel J Math, 1995

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 satisfies the
n-construction principle.

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