The structure of sublattices
of the product of n lattices is explored. Such a sublattice is decomposed and
completely characterized in terms of n(n − 1)∕2 sublattices of the product of two
lattices. A sublattice of the product of two lattices is represented in terms of several
easily characterized sublattices. The sublattice characterizations provide analogous
characterizations for those functions whose level sets are sublattices. A simple
representation is also given for the sections of a sublattice of the product of two
lattices.