Infinitary Axiomatizabiliy of Slender and Cotorsion-free Groups

Oren Kolman & Saharon Shelah

This talk contains some results on the axiomatizability in higher-order logic of the classes of slender and cotorsion-free infinite abelian groups.

Recall that the Baer-Specker group ℤ^{ω} is the product of countably many copies of ℤ (the integers). A torsion-free group G is slender if whenever h : ℤ^{ω} → G is a homomorphism, then h(e_{n}) = 0 for all but finitely many n ∈ ℕ, where e_{n} is the standard n-th unit vector (0, …, 0, 1, 0, …) ∈ ℤ^{ω}. A group A is cotorsion if Ext(J, A) = 0 for all torsion-free groups J. A group G is cotorsion-free if G does not contain any non-zero subgroup which is cotorsion.

In 1974, Paul Eklof used infinitary logic to generalize some classical theorems of infinite abelian group theory. He characterized the strongly ℵ_{1}-free groups as exactly those abelian groups which are L_{∞ω_{1}}- equivalent to free abelian groups, used his criterion to deduce that the class of free abelian 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 that ℤ^{ω} is not free. The present talk continues in the tradition allying infinite abelian group theory with infinitary logic. Its main results are the following:

Theorem 1 The class of slender abelian groups is axiomatizable in the infinitary logic L_{∞ω_{1}}.

Corollary 2 The Baer-Specker group ℤ^{ω} is not L_{∞ω_{1}}-equivalent to a slender group.

Corollary 3 The class of cotorsion-free groups is L_{∞ω}-definable.

Addresses Oren Kolman: King’s College London, Strand, London WC2R 2LS, U.K.

Saharon Shelah: Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel