Potential Methods for Extending Galvin and Jónsson's Characterization of Distributive Sublattices of Free Lattices
arXiv:1603.05124
Abstract
In 1959, F.Galvin and B.Jonsson characterized distributive sublattices of free lattices in their paper. In this paper, I will create new proofs to a portion of Galvin and Jónsson's results. Based on these new proofs, I will explore possible generalizations of F.Galvin and B.Jónsson's work by defining \emph{spanning pairs} and proving partial results which may help with analysing finite width sublattices of free lattices; and by making some new observations on finitely generated lattices over semidistributive varieties. The work done in this paper may assist in attacking the following long-standing open problem: Which countable lattices are isomorphic to a sublattice of a free lattice?