A Wilcox lattice L is
constructed from a complemented modular lattice Λ, by deleting nonzero elements
of some ideal of Λ and by introducing in the remains L the same order as
Λ. The lattice Λ is called the modular extension of L. Using the theory
of parallelism in atomistic lattices, it was proved that any affine matroid
lattice is an atomistic Wilcox lattice, that is, an existence theorem of the
modular extension in the atomistic case. The main purpose of this paper is to
extend this result to the general case, by the use of arguments on point-free
parallelism.
