  In this article, the notion of bi-monotonic independence is introduced as an
extension of monotonic independence to the two-faced framework for a family of
pairs of algebras in a non-commutative space. The associated cumulants are
defined and a moment-cumulant formula is derived in the bi-monotonic setting.
In general the bi-monotonic product of states is not a state and the
bi-monotonic convolution of probability measures on the plane is not a
probability measure. This provides an additional example of how positivity need
not be preserved under conditional bi-free convolutions.
