Abstract
In this paper the study of which varieties, in a countable similarity type, have non-free\(L_{\infty _\omega }\) (or equivalently ℵ1-free) algebras is completed. It was previously known that if a variety satisfies a property known as the construction principle then there are such algebras. If a variety does not satisfy the construction principle then either every\(L_{\infty _\omega }\)-free algebra is free or for every infinite cardinalk, there is a k+-free algebra of cardinality k+ which is not free. Under the set theoretic assumption V=L, for any varietyV in a countable similarity type, either the class of free algebras is definable in\(L_{\omega _1 \omega }\) or it is not definable in any\(L_{\infty _\kappa }\).
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References
P. Eklof, andA. Mekler,Categoricity results for L ∞k -free algebras, Annals of Pure and Applied Logic37 (1988) 81–99.
R.Grossberg and B.Hart,Classification of excellent classes, J. Symbolic Logic (to appear)
B.Hart, Some results in classification theory, Ph.D. Thesis, McGill, 1987.
G. Higman,Almost free groups, Proc. London Math. Soc.1 (1951), 231–248.
W. Hodges, For singular, λ, λ-free implies free, Algebra Universalis12 (1981), 205–220.
A.Mekler, Applications of logic to group theory, Ph.D. thesis, Stanford, 1976.
A.Mekler, Almost free groups in varieties, J. Algebra, (to appear)
S. Shelah,A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math.21 (1975), 319–340.
S.Shelah, Classification Theory, North Holland 1978.
S. Shelah,Classification theory for non-elementary classes I: the number of uncountable models of ψ ε L ω1ω.Part A. Israel J. Math.46 (1983) 211–240.
S. Shelah,Classification theory for non-elementary classes I: the number of uncountable models of ψ ε L ω1ω.Part B. Israel J. Math.46 (1983) 241–273.
S. Shelah,Incompactness in Regular Cardinals, Notre Dame J. Formal Logic26 (1985) 195–228.
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In Memory of Evelyn Nelson
Research partially supported by NSERC of Canada Grant #A8948.
Research partially supported by NSERC of Canada. The research for this paper was begun while the second author was visiting Simon Fraser University.
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Mekler, A.H., Shelah, S. \(L_{\infty _\omega }\)free algebras. Algebra Universalis 26, 351–366 (1989). https://doi.org/10.1007/BF01211842
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DOI: https://doi.org/10.1007/BF01211842