A Simple Sufficient Condition for a Finitely Generated Lattice to not be Embeddable in a Free Lattice

TL;DR

This paper introduces a simple, verifiable condition to determine when finitely generated lattices cannot be embedded in free lattices, advancing understanding of sublattice embeddability and providing new counterexamples.

Contribution

It presents a novel, easily checkable sufficient condition for non-embeddability of finitely generated lattices into free lattices, addressing an open problem in lattice theory.

Findings

Derived a simple sufficient condition for non-embeddability.

Provided counterexamples to a common misconception about finitely generated semidistributive lattices.

Identified a new property of free lattices that may be previously unknown.

Abstract

This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable lattices) are sublattices of a free lattice?" Despite partial progress over the decades, the problem is still unsolved. There is emphasis on the countable case because the current body of knowledge on sublattices of free lattices is most concentrated on when these sublattices are countably infinite. In this article, a simple sufficient condition for a \emph{finitely generated} lattice to not be embeddable in a free lattice is derived, which is easier to verify than checking if Jonsson's condition, L = D(L) = D^d(L), holds. It provides a systematic way of providing counterexamples to the false claim that: "all finitely generated semidistributive lattices satisfying Whitman's condition is a sublattice of a free lattice." To the best of the author's knowledge, this result is new. A corollary that is derived from this is a non-trivial property of free lattices which might not yet be known in the literature. Moreover, a problem concerning finitely generated semidistributive lattices will be posed.