Antichain
Join and meet operations

Any antichain A {\displaystyle A} corresponds to a lower set

In a finite partial order (or more generally a partial order satisfying the ascending chain condition) all lower sets have this form. The union of any two lower sets is another lower set, and the union operation corresponds in this way to a join operationon antichains:

Similarly, we can define a meet operation on antichains, corresponding to the intersection of lower sets:

The join and meet operations on all finite antichains of finite subsets of a set X {\displaystyle X} define a distributive lattice, the free distributive lattice generated by X. {\displaystyle X.} Birkhoff's representation theorem for distributive lattices states that every finite distributive lattice can be represented via join and meet operations on antichains of a finite partial order, or equivalently as union and intersection operations on the lower sets of the partial order.
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