Free and Finitely Presented Lattices

Description: We'll say more later but for now we'll just say that free lattices don't cost much. In fact the book only cost $39.

(R. A. Dean) Can a free lattice have an ascending chain of sublattices all isomorphic to FL(3)?
Which unary polynomials on free lattices are fixed point free? For which unary polynomials f does the join of fⁱ(a) exist for all a?
Characterize those ordered sets which can be embedded into a free lattice.
Which lattices (and in particular which countable lattices) are sublattices of a free lattice?
Does there exist a quadruple a < c₁ <= c₂ < b of elements of a free lattice FL(X) such that the intervals c₁/a and b/c₂ are both infinite and every element of b/a is comparable with either c₁ or c₂?
Describe all meet reducible elements a of FL(X) such that every element above a is comparable with a canonical meetand of a.
Which completely join irreducible elements a in FL(X) are such that kappa(a) is the dual of a? Are there infinitely many for fixed X?

Is there a polynomial time algorithm which decides if an element w in a finitely presented lattice FL(P) is completely join irreducible?
Is there a polynomial time algorithm which decides if a finitely presented lattice is finite?
Is there an algorithm to decide if a finitely presented lattice is weakly atomic? of finite width?
Is there a polynomial time algorithm which decides if a finitely presented lattice is projective?
If L is a finitely presented lattice in which every nonzero join irreducible element has a lower cover and the dual property holds, is L finite?

Does every finitely generated lattice variety have a finite, convergent AC term rewrite system? What about every variety generated by a finite lower (or upper) bounded lattice?
Can one decide if an ordered set of size n is a lattice in time faster than O(n^5/2)?
If d(n) denotes the minimum cardinality of a lattice of order dimension n, is d(n) in O(n²)?
Is there an infinite lattice which is order polynomially complete?