$L_{\infty _\omega }$free algebras

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$L_{\infty _\kappa }$.