Interval hypergraphs and D-interval hypergraphs

A hypergraph H = (V, E) is called an interval hypergraph if there exists a one-to-one function ƒ mapping the elements of V to points on the real line such that for each edge E, there is an interval I, containing the images of all elements of E, but not the images of any elements not in E1. The difference hypergraph D(H) determined by H is formed by adding to E all nonempty sets of the form E1 − E1, where E1 and E1 are edges of H H is said to be a D-interval hypergraph if D(H) is an interval hypergraph. A forbidden subhypergraph characterization of D-interval hypergraphs is given. By relating D-interval hypergraphs to dimension theory for posets, we determine all 3-irreducible posets of length one.