Boolean Hierarchies of Partitions over a Reducible Base

The Boolean hierarchy of partitions was introduced and studied by Kosub and Wagner, primarily over the lattice of $NP$-sets. Here, this hierarchy is treated over lattices with the reduction property, showing that it has a much simpler structure in this instance. A complete characterization is given for the hierarchy over some important lattices, in particular, over the lattices of recursively enumerable sets and of open sets in the Baire space