Two applications of Boolean valued analysis

The paper contains two main results that are obtained by using Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices onto the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of $AL^p$ and $c_o(\Gamma)$ for the class of the so-called $\mathbb{B}$-cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.