Finding bases of uncountable free abelian groups is usually difficult

Additive Bases in Abelian Groups

Let G be a finite, non-trivial abelian group of exponent m, and suppose that B1, . . . , Bk are generating subsets of G. We prove that if k > 2m ln log2 |G|, then the multiset union B1 ∪ · · · ∪ Bk forms an additive basis of G; that is, for every g ∈ G there exist A1 ⊆ B1, . . . , Ak ⊆ Bk such that g = ∑k i=1 ∑ a∈Ai a. This generalizes a result of Alon, Linial, and Meshulam on the additive base...

Intuitionistic Free Abelian Groups

Abelian Groups, Gauss Periods, and Normal Bases

A result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity. Dedicated to Professor Chao Ko on his 90th birthday.

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...

ON FREE PROFINITE GROUPS OF UNCOUNTABLE RANK∗ by