By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to be:
- Is there an equational class between the modular class and the distributive class (different from both)?
Is this question still open?
One could attack it by getting an intimate knowledge of a large family of lattices which would amount in the full classification of that family. The hope would be of finding an intermediate equational class by pointing to a member of the said large family.
First let's mention that $\mathbb R^n$ admits a partial order
$$ x\le y\quad\Leftarrow:\Rightarrow\quad \forall\ _{k=1}^n\ \ x_k\le y_k$$
for every $x\ y\in\mathbb R^n$. Space $\mathbb R^n$ is a distributive lattice with respect to this ordering. Let us consider lattices $L\subseteq \mathbb R^n$ which are lattices with respect to the induced partial order, and which are closed under translations:
$$\forall_{x\in L}\forall_{t\in\mathbb R}\quad x+t\in L$$
where $(x+t)_k:= x_k+t$ for every $k=1\ldots n$.
Call such lattices $L$ translational lattices in $\mathbb R^n$.
REMARK In general, translational lattices in $\mathbb R^n$ are NOT sublattices of $\mathbb R^n$; they are often not distributive, and that's the point. We are searching for a modular non-distributive lattice which satisfies an additional "polynomial" identity not satisfied by some modular lattices.
The goal of a full classification of translation lattices in all spaces $\mathbb R^n$ is realistic. The topic is a mix of combinatorics and geometry (may be with a touch of topology, next to nothing). The translational lattices have rather simple geometric shapes, possible to classify.
DIGRESSION (sorry for my ignorance) Were any other equational classes of lattices studied besides the modular lattices, distributive lattices, and all lattices?
