$M$-free Abelian Groups

On $m^{th}$-autocommutator subgroup of finite abelian groups

Let $G$ be a group and $Aut(G)$ be the group of automorphisms of‎ ‎$G$‎. ‎For any natural‎ number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $G$ is defined‎ ‎as‎: ‎$$K_{m} (G)=langle[g,alpha_{1},ldots,alpha_{m}] |gin G‎,‎alpha_{1},ldots,alpha_{m}in Aut(G)rangle.$$‎ ‎In this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎ ‎all finite abelian groups‎.

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.