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I cannot tell if this will help, but mention it just in case. The area of formal concept analysis deals with algorithms for constructing lattices from sets of objects and attributes. A concept is a tuple $C = (O, A, R)$, where $O$ and $A$ are sets and $R$ is a binary relation from $O$ to $A$. The relation gives rise to a standard function $f$ from the powerset of $O$ to the powerset of $A$.
$f$ maps $X \subseteq O$ to $\{ y \in A \mid \text{for all }x \in X, (x,y) \in R \}$
A function $g$ from the powerset of $A$ to powerset of $O$ is similarly defined such that $f$ and $g$ form a Galois connection. The concept lattice consists of the Galois stable subsets of $A$. By choosing the relation $R$, one can generate lattices with various properties. Algorithms for lattice construction are surveyed in: