  A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and
$B$, consisting of all down-sets in $A \times B$ that are join-closed in either
coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps
from $A$ to $B$, turning arbitrary joins into meets. We introduce more general
kinds of tensor products for closure spaces and for posets. They have the
expected universal property for bimorphisms (separately continuous maps or maps
preserving restricted joins in the two components) into complete lattices. The
appropriate ingredient for quantale constructions is here distributivity at the
bottom, a generalization of pseudo\-complementedness. We show that the
truncated tensor product of a complete lattice $B$ with itself becomes a
quantale with the closure of the relation product as multiplication iff $B$ is
pseudocomplemented, and the tensor product has a unit element iff $B$ is
atomistic. The pseudocomplemented complete lattices form a semicategory in
which the hom-set between two objects is their tensor product. The largest
subcategory of that semicategory has as objects the atomic boolean complete
lattices, which is equivalent to the category of sets and relations. More
general results are obtained for closure spaces and posets.
