Solutions to five problems on tensor products of lattices and related matters. (Q1771930)

The extended tensor product of lattices \(A\) and \(B\) is the lattice of all bi-ideals of \(A\times B\). The tensor product \(A\otimes B\) is defined to be the \(v\)-semilattice of all compact elements of the extended tensor product of \(A\) and \(B\). A capping of a bi-ideal \(I\) of \(A\times B\) is a subset \(\Gamma \) of \(A\times B\) such that \(I\) is the hereditary subset of \(A\times B\) generated by \(\Gamma \;\cup \perp \), where \(\perp\;= (A\times \{0_B\})\cup (\{0_A\}\times B)\). A tensor product is capped if all its elements are capped bi-ideals. A lattice \(A\) with \(0\) is amenable if \(A\otimes L\) is capped for every lattice \(L\) with zero. The author proves among other results: Let \(A\) be a lattice with \(0\). If \(A\otimes L\) is a lattice for every lattice \(L\) with \(0\), then \(A\) is locally finite and \(A\otimes L\) is a capped tensor product. There exists an infinite, three-generated, 2-modular lattice \(K\) with \(0\) such that \(K\otimes K\) is a capped tensor product.