Catalog · 10.5281/zenodo.23477016
Ascending chains of sublattices in free lattices
- Martino Lupini · University of Bologna
19 October 2026 · CC BY 4.0 · latest of 2 versions
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Ascending chains of sublattices in free lattices
Martino Lupini
Abstract
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.
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BibTeX
@unpublished{Lupini202623477016,
title = {Ascending chains of sublattices in free lattices},
author = {Martino Lupini},
year = {2026},
doi = {10.5281/zenodo.23477016},
note = {Zenodo deposit, latest of 2}
}