Tensor Products and Transferability of Semilattices

In general, the tensor product, A ⊗ B, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If A ⊗ B is a capped tensor product, then A ⊗ B is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that A ⊗ B is a capped tensor product, for every lattice B with zero; we shall call such lattices amenable. The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill [5] defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that a finite lattice A is amenable iff it is sharply transferable as a join-semilattice. For a general lattice A with zero, we obtain the result: A is amenable iff A is locally finite and every finite sublattice of A is transferable as a join-semilattice. This yields, for example, that a finite lattice A is amenable iff A⊗F(3) is a lattice iff A satisfies (T), with respect to ∨. In particular, M3 ⊗ F(3) is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.