Covers in free lattices

Freese, Ralph; Nation, J. B.

doi:10.1090/S0002-9947-1985-0773044-4

Covers in free lattices
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by Ralph Freese and J. B. Nation PDF

Trans. Amer. Math. Soc. 288 (1985), 1-42 Request permission

Abstract:

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.

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