  In a recent paper, the authors have proved that for lattices A and B with
zero, the isomorphism $Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds,
provided that the tensor product satis&#64257;es a very natural condition (of
being capped) implying that $A\otimes B$ is a lattice. In general, $A \otimes
B$ is not a lattice; for instance, we proved that $M\_3\otimes F(3)$ is not a
lattice. In this paper, we introduce a new lattice construction, the box
product for arbitrary lattices. The tensor product construction for complete
lattices introduced by G. N. Raney in 1960 and by R. Wille in 1985 and the
tensor product construction of A. Fraser in 1978 for semilattices bear some
formal resemblance to the new construction. For lattices A and B, while their
tensor product $A\otimes B$ (as semilattices) is not always a lattice, the box
product, $A \bp B$, is always a lattice. Furthermore, the box product and some
of its ideals behave like an improved tensor product. For example, if A and B
are lattices with unit, then the isomorphism $Conc(A $\bp$ B) \cong Conc A
\otimes Conc B$ holds. There are analogous results for lattices A and B with
zero and for a bounded lattice A and an arbitrary lattice B. A join-semilattice
S with zero is called {0}-representable, if there exists a lattice L with zero
such that $Conc L \cong S$. The above isomorphism results yield the following
consequence: The tensor product of two {0}-representable semilattices is
{0}-representable.
