Covers in Free Lattices
Ralph S. Freese, J. B. Nation · Transactions of the American Mathematical Society · 1985
In this paper we study the covering relation $(u \succ v)$ in finitely generated free lattices. The basic result is an algorithm which, given an element $w \in {\text {FL}}(X)$, finds all the elements which cover or are covered by $w$ (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in ${\text {FL}}(X)$ are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.