Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Tensor Products and Transferability of Semilattices

Open Access article
 Printed: Aug 1999
  • G. Gr├Ątzer
  • F. Wehrung
Format:   HTML   LaTeX   MathJax   PDF   PostScript  


In general, the tensor product, $A \otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a {\it capped\/} tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for {\it every\/} lattice $B$ with zero; we shall call such lattices {\it amenable}. The first author introduced in 1966 the concept of a {\it sharply transferable lattice}. In 1972, H.~Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that {\it a finite lattice $A$ is\/} amenable {\it if{}f it is\/} sharply transferable {\it as a join-semilattice}. For a general lattice $A$ with zero, we obtain the result: {\it $A$ is amenable if{}f $A$ 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 if{}f $A\otimes\FL(3)$ is a lattice if{}f $A$ satisfies (T), with respect to join. In particular, $M_3\otimes\FL(3)$ 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.
Keywords: tensor product, semilattice, lattice, transferability, minimal pair, capped tensor product, semilattice, lattice, transferability, minimal pair, capped
MSC Classifications: 06B05, 06B15 show english descriptions Structure theory
Representation theory
06B05 - Structure theory
06B15 - Representation theory

© Canadian Mathematical Society, 2018 :