The essentially free spectrum of a variety (Q1912771)

An infinite algebra \(M\) in a variety of algebras is called almost free if it can be represented as the union of an increasing continuous sequence of free subalgebras of size \(< |M |\). It is said to be essentially free if the free product \(M* M'\) is free, for some free algebra \(M'\); otherwise \(M\) is called essentially non-free. The authors study the question in which cardinalities the variety has an almost free, essentially free, non-free member. The first two of the authors [Isr. J. Math. 89, 237-259 (1995; Zbl 0821.08008)] showed that, in general, there is no simple description in ZFC of the class of cardinalities of almost free, essentially non-free algebras in a variety. On the other hand, they conjectured that the spectrum of cardinalities of almost free, essentially free, non-free algebras in an arbitrary variety consists of all successor cardinals, or is empty. Motivating examples for the conjecture are the varieties of abelian groups of exponent 6 and 4 (where the spectra are the class of all successor cardinals and empty, respectively). Towards the conjecture, in the paper under review it is proven that, for an arbitrary variety, (i) each cardinal in the spectrum is a successor, and (ii) the spectrum consists of all successor cardinals provided it contains a cardinal of the form \((\mu^{\aleph_0} )^+\).