More Abelian groups with free duals

More Abelian Groups with Free Duals

In answer to a question of A. Blass, J. Irwin and G. Schlitt, a subgroup G of the additive group Zω is constructed whose dual, Hom(G,Z), is free abelian of rank 2א0 . The question of whether Zω has subgroups whose duals are free of still higher rank is discussed, and some further classes of subgroups of Zω are noted.

Intuitionistic Free Abelian Groups

Free abelian lattice-ordered groups

Let n be a positive integer and FA`(n) be the free abelian latticeordered group on n generators. We prove that FA`(m) and FA`(n) do not satisfy the same first-order sentences in the language L={+,−, 0,∧,∨} if m 6= n. We also show that Th(FA`(n)) is decidable iff n ∈ {1, 2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices. A. M. S. C...

Rainbow-free 3-colorings in abelian groups

A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural description proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.

Some ugly א1-free abelian groups

Given an א1-free abelian group G we characterize the class CG of all torsion abelian groups T satisfying Ext(G, T ) = 0 assuming the special continuum hypothesis CH. Moreover, in Gödel’s constructable universe we prove that this characterizes CG for arbitrary torsion-free abelian G. It follows that there exist some ugly א1-free abelian groups.