  We introduce and explore the concept of positive ideals for both linear and
multilinear operators between Banach lattices. This paper delineates the
fundamental principles of these new classes and provides techniques for
constructing positive multi-ideals from given positive ideals. Furthermore, we
present an example of a positive multi-ideal by introducing a new class,
referred to as positive (p1,...,pm;r)-dominated multilinear operators. We
establish a natural analogue of the Pietsch domination theorem and Kwapien's
factorization theorem within this class.
