Abstract view
Tensor Products and Transferability of Semilattices


Published:19990801
Printed: Aug 1999
Abstract
In general, the tensor product, $A \otimes B$, of the lattices $A$ and
$B$ with zero is not a lattice (it is only a joinsemilattice 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 joinsemilattice}.
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 joinsemilattice}.
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
