α-cut-complete Boolean algebras

Let A be a Boolean algebra, and cc an infinite cardinal number or the symbol co. An cc-cut in A is an ordered pair (F, H) of subsets of A, each of power < alpha, with F less than or equal to H elementwise, with 0 as the meet of differences h -f (h is an element of H, f is an element of F). A is called alpha-cut-complete if for each alpha-cut (F, H) there is alpha is an element of A with F less than or equal to a less than or equal to H elementwise. We describe the simply-constructed alpha-cut-completion A(alpha), show that alpha-cut-completeness solves a natural alpha-injectivity problem, determine when A(alpha) is the alpha-completion, or the completion, and interpret most of that topologically in Stone spaces. Oddly, these considerations seem novel in Boolean algebras, while for lattice-ordered groups and vector lattices, and dually for topological spaces, the analogous theory, especially for alpha = omega(1), has received considerable study.