Embeddings into $\mathbb{B}$-cyclic Banach spaces

For a complete Boolean algebra $\mathbb{B}$ and nonzero $\pi\in \mathbb{B}$, the notion of an $\mathbb{B}_{\pi}$-embedding of Banach spaces into $\mathbb{B}$-cyclic Banach spaces is introduced. The notion of a lattice $\mathbb{B}_{\pi}$-embedding of Banach lattices into $\mathbb{B}$-cyclic Banach lattices is also introduced. A criterion for the $\mathbb{B}_{\pi}$-embedding of a space of conti-\eject nuous vector-valued functions with values in an arbitrary Banach space into a $\mathbb{B}$-cyclic Banach space is established, as well as a criterion for the lattice $\mathbb{B}_{\pi}$-embedding of a space of continuous vector-valued functions with values in an arbitrary Banach lattice into a $\mathbb{B}$-cyclic Banach lattice. The obtained results allow us to outline an approach for isometric and isomorphic classification of $\mathbb{B}$-cyclic Banach spaces. In the course of establishing the results, the tool of lattice-valued spaces was widely used.