  After collecting a number of results on interval and almost interval
preserving linear maps and vector lattice homomorphisms, we show that direct
systems in various categories of normed vector lattices and Banach lattices
have direct limits, and that these coincide with direct limits of the systems
in naturally associated other categories. For those categories where the
general constructions do not work to establish the existence of general direct
limits, we describe the basic structure of those direct limits that do exist. A
direct system in the category of Banach lattices and contractive almost
interval preserving vector lattice homomorphisms has a direct limit. When the
Banach lattices in the system all have order continuous norms, then so does the
Banach lattice in a direct limit. This is used to show that a Banach function
space over a locally compact Hausdorff space has an order continuous norm when
the topologies on all compact subsets are metrisable and (the images of) the
continuous compactly supported functions are dense.
