Infinite intervals in free lattices
The subject of free lattices has long been of interest in lattice theory, and the determination of their structure involves some deep and subtle proof techniques. Considerable progress was made in the landmark paper by \textit{R. Freese} and \textit{J. B. Nation} [Trans. Am. Math. Soc. 288, 1-42 (1985; Zbl 0567.06008)], and the current paper was inspired by a question posed by these authors: ``Does every infinite interval in the free lattice FL(X) generated by X have FL(3) as a sublattice?'' This question is answered in the affirmative by proving that for any set Y with {\#}Y\(\leq \#X+\aleph_ 0\), every infinite interval of FL(X) has a sublattice isomorphic to FL(Y). The paper is written in a clear concise manner, explains the underlying issues very nicely, and closes with a discussion of some related open questions.
scientific article; zbMATH DE number 4162948