Boolean powers of abelian groups (Q750625)

Let A be an abelian group, B a complete Boolean algebra. The Boolean power \(A^{(B)}\) comprises all f: \(A\to B\) with \(\bigvee_{u\in A}f(u)=1\) and \(f(u)\wedge f(v)=0\) for \(u\neq v\). The first observation is that \(A^{(B)}\) is cotorsion-free if A is. Let \(C^{\kappa}\) be the complete Boolean algebra of the regular open sets of \(\kappa^{\omega}\). Here \(\kappa^{\omega}\) is endowed with the product topology of the discrete topological space \(\kappa\). The Main Theorem shows the equivalence of the following three statements. 1) A is cotorsion-free, 2) \(Hom({\mathbb{Z}}^{(C^{\kappa})},A)=0\) for some ordinal \(\kappa\). 3) There is a \(\kappa\) with \(Hom({\mathbb{Z}}^{(C^{\lambda})},A)=0\) for all \(\lambda\geq \kappa\). This theorem fits to the notion of strong cotorsion- free groups, cf. \textit{M. Dugas} and \textit{R. Göbel} [Pac. J. Math. 118, 79-104 (1985; Zbl 0578.20050)], i.e. groups A for which \(Hom({\mathbb{Z}}^{\kappa}/{\mathbb{Z}}^{<\kappa},A)=0\) for all regular cardinals \(\aleph_ 1\leq \kappa <\aleph_ m\) \((=\) first measurable cardinal) and also shows the difference to cotorsion-free groups. Note that Boolean powers can be viewed as a generalization of the quotient \({\mathbb{Z}}^{\kappa}/{\mathbb{Z}}^{<\kappa}.\) Additionally the relation of Boolean powers to algebraically compact groups is described. Let \(\kappa\) be an infinite cardinal, \(A_{\kappa}={\tilde {\mathbb{Z}}}^{\kappa}/\oplus_{\kappa}{\mathbb{Z}}\), B a complete Boolean algebra, then \(A_{\kappa}^{(B)}\) is algebraically compact if and only if B is (\(\omega\),\(\kappa\))-distributive. Here \({\tilde {\mathbb{Z}}}^{\kappa}\) denotes all elements of \({\mathbb{Z}}^{\kappa}\) with countable support.