Two representations of finite ordered sets (Q1343248)

A partitive hypergraph is a pair \((X,E)\), where \(X\) is a finite set and \(E\) is a family of subsets of \(X\), containing \(\emptyset\) and \(X\), which is closed under intersection and for properly intersecting subsets is closed under join and symmetric difference. A partitive lattice is a lattice isomorphic to a partitive hypergraph. The atomic extension of an ordered set \(T_ 1\) by an ordered set \(T_ 2\) in an atom \(a\in T_ 1\) is a substitution of the interval \([0, a]\) by \(T_ 2\) in \(T_ 1\). The author describes a representation of a partitive lattice by atomic extensions for arbitrary finite ordered sets. Moreover, he gives another representation using a system of elementary ordered sets directed by a lattice tree.