In this paper, the authors define
the orde r sum of a family of distributive lattices which is indexed by a partially
ordered set P. The order sum reduces to the free product when P is trivially ordered,
and to the ordinal sum when P is simply ordered.
It is proved that the order sum of conditionally implicative lattices is
conditionally implicative, and that every projective distributive lattice is
conditionally implicative. The second half of the paper investigates conditions under
which the order sum of projective lattices is projective. It is shown that if
{Lα|α ∈ P} is a family of distributive lattices having largest and smallest elements,
then the order sum of the family is projective if and only if each Lα is projective, and
P is such that the order sum of the family {Mα|α ∈ P} of one-element lattices Mα is
projective.
