I am given a (smallish, say $n=14$ element) set $X$, and a set $R$ of (a few hundred) quadruples of elements $(a, b, c, d)$ with $a,b,c, d\in X$.
I want to construct lattices on $X$, such that for all incomparable $a, b\in X$ the quadruple $(a,b,a\vee b, a\wedge b)$ is in $R$.
A trivial solution is to use an $n$-element chain as the lattice, which already gives me $14!=$a huge number of lattices. Thus, in a first step I'd like to find all non-isomorphic lattices. (Aside: is there a lattice structure on the set of isomorphism types of lattices on $n$ elements?)
Currently, I do not have a clue how many lattices I could build with my set, but I am certainly more interested in lattices with many incomparable elements.
As a first step, I determined which pairs $a,b$ do not occur as first two elements in one of the quadruples in $R$, because these must be comparable. In the first case I'm interested in, there are a dozen of these.
How could I do an exhaustive search?