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?