  Several recent papers were devoted to various modifications of limited,
Grothendieck, and Dunford--Pettis operators, etc., through involving the Banach
lattice structure. In the present paper, it is shown that many of these
operators appear as operators affiliated to well known properties of Banach
lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck
property, the property (d), and the sequential w*-continuity of the lattice
operations. It is proved that the spaces consisting of regularly versions of
the above operators are all Banach spaces. The domination problem for such
operators is investigated.
