  We study completions of Archimedean vector lattices relative to any nonempty
set of positively-homogeneous functions on finite-dimensional real vector
spaces. Examples of such completions include square mean closed and geometric
closed vector lattices, amongst others. These functional completions also lead
to a universal definition of the complexification of any Archimedean vector
lattice and a theory of tensor products and powers of complex vector lattices
in a companion paper.
