This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says that for a finite distributive lattice $L$ with set of join-irreducible elements $P$ the subsets $\{p\in P|p\le a\}\subset P, a\in L$ form the set of order ideals of a partial order on $P$ (which uniquely defines the partial order). The latter says that if $L$ is, instead, a geometric lattice, then the same family of subsets is the set of flats for a matroid structure on $P$ (for a geometric lattice join-irreducible elements are precisely its atoms). The former provides a bijection between finite distributive lattices and finite posets, the latter provides a bijection between geometric lattices and finite simple matroids. Both bijections have functorial interpretations.

Broadly speaking, my question is: are there other interesting correspondences of this form and do these two phenomena have some interesting common generalization? More specifically, here is an example of what such a result may look like. We may consider an arbitrary (finite?) lattice $L$ with set of join-irreducible elements $P$ and define a family of subsets of $P$ in the same way. Does this family of subsets define some interesting algebraic structure on $P$? Perhaps, for some specific classes of lattices, maybe classes that include both geometric and distributive $L$? (Apparently, in full generality we can obtain any family which is closed under intersection, has least common supersets and for any $p,q\in P$ includes a subset containing exactly one of $p$ and $q$. But I'm not sure where to go from here.)

Update. Two further nice examples of such correspondences were given by Richard Stanley and Sam Hopkins.

• For finite join-distributive $L$ this family of subsets is the family of feasible sets of an antimatroid on $P$. This generalizes Birkhoff's representation theorem. See details here.
• In their 2019 paper Reading, Speyer and Thomas introduce the notion of a finite two-acyclic factorization system: a finite set equipped with a specific kind of binary relation. For a finite semidistributive lattice our family of subsets is the set of first components of maximal orthogonal pairs of such a binary relation on $P$ and the relation is recovered from this data.

I've accepted the first answer but I still very much hope to see other examples!