On D-complementation (Q798326)

In a distributive lattice D, an order isomorphism \(\gamma\) is given from the join-irreducible elements to the meet-irreducible elements, and is extended to a bijection of D, called D-complementation. Some examples and results are given in the case where D is D(P), the set of descending subsets (or, equivalently, antichains) of a partially ordered set P. The results are then used to define a generalization of a matroid on a poset, in which the map \(\gamma\) in D(D(P)) gives the bases from the circuits and the hyperplanes from the bases.