A note on well-generated Boolean algebras in models satisfying Martinʹs axiom Original Research Article

A Boolean algebra B is well generated, if it has a well-founded sublattice L such that L generates B. Let B be a superatomic Boolean algebra. The rank of B (image) is defined to be the Cantor Bendixon rank of the Stone space X of B. For every image let image be the number of isolated points in the iʹs Cantor Bendixon derivative of X. The cardinal sequence of B is defined as image. If image, then the rank of a (image) is defined as the rank of the Boolean algebra image. An element image is a generalized atom (image), if the last cardinal in the cardinal sequence of image is 1. Let image. We denote image, if image. A subset image is a complete set of representatives (CSR) for B, if for every image there is a unique image such that image. Any CSR for B generates B. We say that B is hereditarily decreasingly canonically well generated, if for every subalgebra C of B and every CSR H for C there is a CSR M for C such that: (1) for every image and image: if image then image; (2) the sublattice of C generated by M is well founded.