Completeness of Boolean powers of Boolean algebras
For complete Boolean algebras A and B, let A[B] denote the Boolean power of A by B. This paper investigates conditions under which A[B] is a complete Boolean algebra, using techniques of Boolean valued models. The link between this problem and Boolean valued models is provided by a result of \textit{R. M. Solovay} and \textit{S. Tennenbaum} [Ann. Math., II. Ser. 94, 201-245 (1971; Zbl 0244.02023)]: A[B] is complete if and only if \(\check A\) is \(complete^{(B)}=1\) in the Boolean valued universe \(V^{(B)}\), where Ǎ is the element of \(V^{(B)}\) that corresponds to A in the standard universe. The results are stated in terms of B, or more precisely, the Boolean valued universe \(V^{(B)}\). Two interesting results from the paper are: (1) for a regular cardinal \(\kappa\), \(\check A\) is \(complete^{(B)}=1\) for every \(\kappa\)-saturated complete Boolean algebra A if and only if B satisfies the \((<\kappa,\infty)\)- distributive law and \([[ \check A\) is \({\check \kappa}\)- saturated\(]]^{(B)}=1\) for every \(\kappa\)-saturated A; (2) if \(\kappa\) is a regular cardinal such that \(2^{<\kappa}=\kappa\), then the following conditions are equivalent: (a) B is \(\kappa\)-representable, (b) B satisfies the \((<\kappa,\kappa)\)-distributive law and \([[ \check A\) is \({\check \kappa}\)- saturated\(]]^{(B)}=1\) for every \(\kappa\)-saturated Boolean algebra A with \(| A| \leq \kappa\), (c) \(\check A\) is \(complete^{(B)}=1\) for every \(\kappa\)-saturated complete Boolean algebra A with \(| A| \leq \kappa\).
scientific article; zbMATH DE number 4070978