Lectures on Free Lattices

Abstract

These lectures notes present the theory of free lattices, assuming only a basic understanding of lattice theory. They begin with Whitman’s solution to the word problem and his canonical form. The well known consequences of these are given as well as several lesser known consequences, such as the continuity of free lattices, the existence of a fixed point free unary polynomial on a free lattice, and the fact that finite sub-lattices of a free lattice satisfy a nontrivial lattice equation. The theory of covers in free lattices is developed and some of the consequences explored. Tschantz’s Theorem and a new characterization of semisingular elements are discussed and some important consequences of these results are given such as the existence of dense maximal chains in intervals of a free lattice.