| Distributive lattices---alone, or with enriched structure---are mathematical objects of fundamental importance. This text studies generalized distributive lattices; the generalization is that certain infinite meets and joins are required to exist. Subset systems (natural rules which select a family of subsets of each poset) j and m label which sets have joins and meets, respectively.; A calculus of subfunctors is developed: using this calculus, it is shown that any subfunctor F of a monad (containing the image of the unit) generates a submonad F¯. Under suitable conditions, any partial F¯-algebra extends to an F¯ -algebra. The monad F¯ for the free distributive (j, m)-complete lattice is the submonad of the completely distributive complete lattice monad generated by a subfunctor obtained from j and m.; The category DPjm of (j, m)-complete lattices which can be embedded in a completely distributive complete lattice is a full subcategory of F¯-algebras. DPjm is complete and has coequalizers.; (j, m)-complete families of subsets of a set (generalized topological spaces) are investigated in analogy to classical point-set topology. Assuming suitable restrictions on j and m, subspaces can be defined. Assuming these restrictions, there are well-behaved categories corresponding to T0- and sober-spaces. |
