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Antimatroids are a good example. We have the syllogism "Antimatroids are to matroids as join-distributive lattices are to geometric lattices." Two other examples are the characterizations of complemented modular lattices of finite length $n\geq 4$ and primary modular lattices of finite length $n\geq 4$. (A modular lattice of finite length is primary if for every join-irreducible $t$, the interval $[\hat{0},t]$ is a chain, and dually, e.g., the lattice of subgroups of a finite abelian group.) Their characterizations are analagous to characterizing finite distributive lattices as a collection of sets closed under union and intersection. See for instance Theorems 5 and 6 of Alan Day, Geometric applications in modular lattices, in Universal Algebra and Lattice Theory (Puebla, 1982), Springer Lecture Notes in Mathematics 1004, pp. 111-141. I don't know a common generalization of distributive and geometric lattices with the type of structure asked for.