Monday, November 29th, 2004
2:30pm - 3:30pm, in Science 2-065

Jonathan Farley

MIT

Tensor Products of Semilattices, Semimodularity and Supersolvability: A Problem of E. T. Schmidt from 1974 and Some Conjectures of Quackenbush from 1985

Abstract: If $M$ is a finite complemented modular lattice with $n$ atoms and $D$ is a bounded distributive lattice, then the Priestley power $M[D]$ is shown to be isomorphic to the poset of so-called normal elements of $D^n$, thus solving a problem of E. T. Schmidt from 1974. It is shown that there exist a finite modular lattice $A$ not having $M_4$ as a sublattice and a finite modular lattice $B$ such that $A\otimes B$ is not semimodular, thus refuting a conjecture of Quackenbush from 1985. It is shown that the tensor product of $M_3$ with a finite modular lattice $B$ is supersolvable if and only if $B$ is distributive, thus proving a conjecture of Quackenbush from 1985.