Ascending chains of sublattices in free lattices
Description
We prove that every increasing sequence of free sublattices of bounded finite rank in a free lattice is eventually constant. In particular, there is no infinite strictly increasing chain of copies of the free lattice on three generators, answering negatively a question of R. A. Dean recorded in 1997. Freeness is essential: we construct in the three-generated free lattice an infinite strictly increasing chain of pairwise nonisomorphic retracts, each requiring exactly three generators. The corresponding retractions commute. We also give a proper self-embedding whose fixed-point sublattice contains a free copy of the ambient lattice, and prove that the stable image of every endomorphism of a finitely generated free lattice consists precisely of its periodic points.