A Sufficient Condition for J\'onsson's Conjecture and its Relationship with Finite Semidistributive lattices

TL;DR

This paper establishes a new sufficient condition involving minimal pairs that implies Jönsson's conjecture, and explores its connection with finite semidistributive lattices, providing new insights into sublattices of free lattices.

Contribution

It proves a new sufficient condition involving minimal pairs for Jönsson's conjecture and investigates its relation to finite semidistributive lattices, offering novel results.

Findings

Proved a sufficient condition involving minimal pairs for Jönsson's conjecture.

Refuted a main assertion of Muhle's manuscript using this condition.

Made a partial result on forbidden sublattice characterization in semidistributive lattices.

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. It is known that sublattices of free lattices which are finite can be characterized as being those lattices which satisfy Whitman's condition and are semidistributive. This assertion was conjectured by B. Jonsson in the 1960's and proven by J.B. Nation in 1980. However, there is a desire for a new proof to this deep result as Nation's proof is very involved and more insight into sublattices of free lattices is sought after. In this article, a sufficient condition involving a construct known as a join minimal pair, or just a minimal pair, implying J'onsson's conjecture is proven. Minimal pairs were first defined by H. Gaskill when analysing sharply transferable lattices. Using this sufficient condition, research by I.Rival and B.Sands is used to compare this condition with properties of finite semidistributive lattices and in the process refute the main assertion of a manuscript by H.Muhle. Moreover, inspired by the approaches used by Henri Muhle, I will make a partial result which investigates a possible forbidden sublattice characterization involving breadth-two planar semidistributive lattices. To the best of my knowledge, the two results of this article (the sufficient condition for Jonsson's conjecture and the partial result aforementioned) are new.