I see from this answer "if you want an algebraic definition, you have to stick with equations.", and this comment, that equations are preferred for defining lattice properties. But it seems to me there's plenty of (useful) properties can't be expressed as equations.
Every element has a complement.
all x. exists z. (x ^ z = 0 & x v z = 1);(How to eliminate the existential quant?)
Every element has a unique pseudo-complement
x* = max{ y ∈ L | x ∧ y = 0 };(Now to eliminate the nested quantifying over 'candidate' complements?)
The Burris and Sankappanavar textbook just doesn't give formulae if it can't give an equation. That's not helpful. (For example Definition 8.6/page 156 for 'relatively complemented'.)
In my specific example, I'm working with a family of lattices which (I think) are 'relatively pseudo-complemented'. That is, within every convex sublattice, every element has a unique max relative complement. How do I express that equationally? (Currently I have a bunch of clunky formulae with existential quant, nested foralls and implications.)