Solutions to five problems on tensor products of lattices and related matters

Abstract

The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the infinite case. In this paper, we answer several open questions about tensor products of lattices. Among the results that we obtain are the following. Theorem 2. Let A be a lattice with zero. If $A \otimes L$ is a lattice for every lattice L with zero, then A is locally finite and $A \otimes L$ is a capped tensor product for every lattice L with zero. Theorem 5. There exists an infinite, three-generated, 2-modular lattice K with zero such that $K \otimes K$ is a capped tensor product. Here, 2-modularity is a weaker identity than modularity, introduced earlier by G. Grätzer and the author.