Categoricity results for L∞κ

Abstract

Let V denote a variety of algebras in a countable language. An algebra is said to be L ∞ κ -free if it is L ∞ κ -equivalent to a ( V-) free algebra. If every L ∞ ω1 -free algebra of cardinality ω 1 is free, then for all infinite cardinals κ every L ∞ κ -free algebra of cardinality κ is free. Further, assuming suitable set-theoretic hypotheses, if there is a non-free L ∞ ω1 -free algebra of cardinality ω 1, then for a proper class of cardinals κ there are non-free L ∞ κ -free algebras of cardinality κ. The varieties in which the class of free algebras are definable by a sentence in L ω1 ω are characterized as the weak Schreier varieties in which every L ∞ ω -free algebra of cardinality ω 1 is free. A weak Schreier variety is one in which every L ∞ ω -elementary substructure of a free algebra is free. In fact, assuming suitable set-theortic hypotheses, for weak Schreier varieties the class of free algebras is definable in L ∞∞ iff it is definable in L ω1 ω . Varieties in uncountable languages are also considered.