A duality for finite lattices

An OD-graph is a triple $(J,\leq,M)$ with $(J,\leq)$ a finite poset and, for each $j$ in $J, M(j)$ a collection of subsets of $J$; these data are subject to additional constraints. Given an OD-graph we can define closed subsets of $J$, whence a finite lattice. Given a finite lattice $L$, we can define its OD-graph $(J(L),\leq,M)$ where $J(L)$ is the set of join-irreducible elements of $L, \leq$ is the restriction of the order to join-irreducible elements, and $C$ in $M(j)$ if and only if $C$ is a minimal join-cover of $j$. By defining morphisms, I'll explain how OD-graphs can be cast into a category. This category turns out to be dual of the category of finite  lattices, with the two constructions described above being the object  part of the contravariant functors that give rise to the duality. I'll sketch how some lattice theoretic properties correspond to first order properties of OD-graphs. That is, for each finite tree $T$, I'll  construct an equation $e_T$ that holds in a finite lattice if and  only its OD-graph does not have some kind of shape from $T$. I'll exemplify the use of the concept presented by discussing the equational theory of the lattices of permutations and the equational theory of extremal fixed points on finite lattices.