  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.
