A non-capped tensor product of lattices

Abstract

In lattice theory, the tensor product ${A \otimes B}$ is naturally defined on (${\vee}$, 0)-semilattices. In general, when restricted to lattices, this construction will not yield a lattice. However, if the tensor product ${A \otimes B}$ is capped, then ${A \otimes B}$ is a lattice. Whether the converse is true is an open problem, first posed by G. Grätzer and F. Wehrung in 2000. In the present paper, we prove that it is not so, that is, there are bounded lattices A and B such that ${A \otimes B}$ is not capped, but is a lattice. Furthermore, A has length three and is generated by a nine-element set of atoms, while B is the dual lattice of A.