Infinitary axiomatizability of slender and cotorsion-free groups

Abstract

AbstractThe classes of slender and cotorsion-free abelian groups are axiomatizable in the infinitary logics L ∞ω 1 and L ∞ω respectively. The Baer-Specker group Z ω is not L ∞ω 1 -equivalent to a slender group. 1 INTRODUCTION In 1974, Paul Eklof [4] used infinitary logic to generalize some classical theorems of infinite abeliangroup theory. He characterized the strongly ℵ 1 -free groups as exactly those abelian groups whichare L ∞ω 1 -equivalent to free abelian groups, used his criterion to deduce that the class of freeabelian groups is not L ∞ω 1 -definable, and showed that the Baer-Specker group Z ω is not L ∞ω 1 -equivalent to a free abelian group, strengthening a theorem of Baer [1] that Z ω is not free. Thispaper continues in the tradition allying infinite abelian group theory with infinitary logic. Itsmain result is the following:Theorem 1.1 The class SL of slender abelian groups is axiomatizable in the infinitary logicL ∞ω 1 .Corollary 1.2 The Baer-Specker group Z ω is not L ∞ω 1 -equivalent to a slender group.Corollary 1.2 improves further the above-mentioned results of Baer [1] and Eklof [4]. Theorem1.1 contrasts strikingly with another of Eklof’s corollaries that the class of free groups is not