I asked:
> > is it known if the quasi-variety of balanced
> > lattices with 0 is actually a variety?

Keith replied:
> It is not a variety. If you add a new zero element to any
> lattice L with zero it becomes balanced, and L is a quotient
> of this new balanced lattice. Thus, the variety generated by
> balanced lattices is the variety of all lattices with zero,
> not all of which are balanced.

Oh, right. In fact, you even answered my next question,
which was about the variety generated by balanced lattices.
Yes, of course. I was just being dense. Thanks, Keith.

To redeem myself, I should ask a different, more
general question: are there any well-known open problems
in lattice theory of the form

"Is the quasi-variety X of lattices actually a variety?"

The motivation for my original question and this one is
that I'm developing an automated reasoning technique that
seems to be useful for questions of this type. I was
playing with the quasi-variety of balanced lattices and
wondering why there didn't seem to be any equational
consequences of the balanced quasi-identity which were not
already true in all lattices with zero. I should have just
thought about it for a few minutes instead of treating it
formally.

MK