This question is motivated by the superficial observation that [Birkhoff's representation theorem](https://en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem) and the [cryptomorphism between matroids and geometric lattices](https://en.wikipedia.org/wiki/Geometric_lattice#Cryptomorphism) 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](https://en.wikipedia.org/wiki/Antimatroid) on $P$. This generalizes Birkhoff's representation theorem. See details [here](https://en.wikipedia.org/wiki/Antimatroid#Join-distributive_lattices).
* In their [2019 paper](https://arxiv.org/abs/1907.08050) 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!**



Birkhoff's representation theorem vs matroid-geometric lattice correspondence
[Antimatroids](https://en.wikipedia.org/wiki/Antimatroid)
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 analogous 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.


