Tensor products of semilattices with zero, revisited

Let A and B be lattices with zero. The classical tensor product, A B, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very natural condition: A B is capped (that is, every element is a finite union of pure tensors) under which the tensor product is always a lattice. Let Conc L denote the join-semilattice with zero of compact congruences of a lattice L. Our main result is that the following isomorphism holds for any capped tensor product: Conc A Conc B Conc(A B).This generalizes from finite lattices to arbitrary lattices the main result of a joint paper by the first author, H. Lakser, and R.W. Quackenbush.