The Edge Covering Number of Ordered Sets

The edge covering number \(e(P)\) of an ordered set \(P\) is the minimum number of suborders of \(P\) of dimension at most two so that every covering edge of \(P\) is included in one of the suborders. Unlike other familiar decompositions, we can reconstruct the ordered set \(P\) from its components. In this paper, we find some familiar ordered sets of edge covering number two and then show that \(e(2^n) \to \infty\) as \(n\) gets large.