Relation between Lattice and Boolean Algebra
I would hardly describe a lattice as a generalized form of boolean algebra, since there are many more things that a lattice can describe.

A better description would be to say that boolean algebra forms an extremely simple lattice. It has two elements, $\top$ and $\bot$, with $\bot \sqsubset \top$. The meet corresponds to conjunction (AND), and the join corresponds to disjunction (OR), though you can make a dual lattice with these flipped.

But, we don't teach this with boolean algebra for the same reason we don't teach group theory when teaching children about the whole numbers.

Lattice theory is way more than you need to understand booleans, and while it's a neat fact that booleans form a simple lattice, the mechanics of lattice theory, fixpoints, continuity, etc. were made to solve much harder problems than arise with booleans.