  In this paper, the notion of bi-Boolean independence for non-unital pairs of
algebras is introduced thereby extending the notion of Boolean independence to
pairs of algebras. The notion of B-$(\ell, r)$-cumulants is defined via a
bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions,
and it is demonstrated that bi-Boolean independence is equivalent to the
vanishing of mixed B-$(\ell, r)$-cumulants. Furthermore, some of the simplest
bi-Boolean convolutions are considered, and a bi-Boolean partial
$\eta$-transform is constructed for the study of limit theorems and infinite
divisibility with respect to the additive bi-Boolean convolution. In
particular, a bi-Boolean L\'{e}vy-Hin\v{c}in formula is derived in perfect
analogy with the bi-free case, and some Bercovici-Pata type bijections are
provided. Additional topics considered include the additive bi-Fermi
convolution, some relations between the $(\ell, r)$- and B-$(\ell,
r)$-cumulants, and bi-Boolean independence in an amalgamated setting.
  The last section of this paper also includes an errata that will be published
with this copy of the paper.
