On Boolean powers of the group Z and (\(\omega\) ,\(\omega\) )-weak distributivity
Let B be a countably complete Boolean algebra (ccBa). Then the Boolean power \({\mathbb{Z}}^{(B)}\) consists of all maps \(x: {\mathbb{Z}}\to B\) such that \(\{\) x(a)\(| a\in {\mathbb{Z}}\}\) forms a partition of unity. A ''product- like'' abelian group structure is obtained by defining \((x+y)(a)=\bigvee_{b\in {\mathbb{Z}}}x(b)\wedge y(a-b),\) in particular \({\mathbb{Z}}^{({\mathcal P}(J))}\cong {\mathbb{Z}}^ J\). B is said to have the slender property if every homomorphism from \({\mathbb{Z}}^{{\mathbb{N}}}\) to \({\mathbb{Z}}^{(B)}\) is ''infinitely linear''. The first author observed [in J. Algebra 82, 84-93 (1983; Zbl 0538.20027)] that (\(\omega\),\(\omega)\)-weakly distributive complete Boolean algebras have the slender property. They extend this result to a ccBa. If I is a countably complete ideal of a ccBa B, then the infinite linearity of \(h: {\mathbb{Z}}^{{\mathbb{N}}}\to {\mathbb{Z}}^{(B/I)}\) is shown to be equivalent with the existence of a lifting homomorphism \(\bar h:\) \({\mathbb{Z}}^{{\mathbb{N}}}\to {\mathbb{Z}}^{(B)}\). This criterion is used to decide the question if the \(ccBa\quad B={\mathcal P}(J)/{\mathcal P}_{\kappa}(J)\) has the slender property, where \({\mathcal P}_{\kappa}(J)\) consists of all subsets of J with cardinality less than \(\kappa\), and \(\kappa\) is a cardinal of uncountable cofinality. With the help of a result by \textit{B. Wald} [Manuscr. Math. 42, 297-309 (1983; Zbl 0518.20052)] in the presence of Martin's axiom the answer is proven to be ''yes'' if and only if cf \(\kappa\) \(\neq 2^{\aleph_ 0}\). Finally using a suitable Boolean extension of the universe the independence of the statement ''\({\mathcal P}(J)/{\mathcal P}_{2^{\aleph_{0}}}(J)\) has the slender property'' is established.
(\(\omega \) ,\(\omega \) )-weakly distributive complete Boolean algebras
scientific article; zbMATH DE number 3886143