An Analysis of Countable Sublattices of Free Lattices

Abstract

This thesis investigates which lattices are countable sublattices of free lattices. This is equivalent to determining which lattices are sublattices of the free lattice on three generators. Our investigation is based on the following open problem (see \cite{FL}, 1995): ``Which lattices (and in particular which countable lattices) are sublattices of a free lattice?'' We describe how stronger variants of the semidistributive laws have contributed to our current understanding of sublattices of free lattices. In particular, we see how the strongest results known only apply to certain classes of countables lattices. Afterwards, we propose a strategy for analyzing the following class of countable lattices: finite width sublattices, L, of free lattices, where L is in a semidistributive variety.