Alan H. Mekler

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 ω1 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 ω1, 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 Lω1ω are characterized as the weak Schreier varieties in which every L∞ω-free algebra of cardinality ω1 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 Lω1ω. Varieties in uncountable languages are also considered.

Some universal graphs

It is shown that various classes of graphs have universal elements. In particular, for eachn the class of graphs omitting all paths of lengthn and the class of graphs omitting all circuits of length at leastn possess universal elements in all infinite powers.

The consistency strength of “every stationary set reflects”

The consistency strength of a regular cardinal so that every stationary set reflects is the same as that of a regular cardinal with a normal idealI so that everyI-positive set reflects in aI-positive set. We call such a cardinal areflection cardinal and such an ideal areflection ideal. The consistency strength is also the same as the existence of a regular cardinal κ so that every κ-free (abelian) group is κ+-free. In L, the first reflection cardinal is greater than the first greatly Mahlo cardinal and less than the first weakly compact cardinal (if any).

\(L_{\infty _\omega }\)free algebras

AbstractIn this paper the study of which varieties, in a countable similarity type, have non-free

Abelian p-groups with no invariants☆

L_{\infty _\omega }

Stationary logic and its friends. II

(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

Totally Crawley groups

L_{\infty _\omega }