A Bruhat order for bipartite graphs whose node sets are posets: Lifting, switching, and adding edges (Q912137)

For finite posets P and Q a deltoid is a bipartite multigraph with edges between P and Q. From a deltoid D one can obtain another deltoid \(D'\) by elementary operations of the following types: 1) Let \(p<p'\) in P and pq an edge in D. Delete pq and add the edge \(p'q.\) 2) Let \(q<q'\) in Q and pq an edge in D. Delete pq and add the edge \(pq'\). 3) Let \(p<p'\) in P, \(q<q'\) in Q and \(pq'\), \(p'q\) edges in D. Delete \(pq'\), \(p'q\) and add pq, \(p'q'.\) 4) Add an edge pq. Theorem. Let L be a linear and Q a series-parallel partial order and let D and \(D'\) be two deltoids between L and Q. Then the following statements are equivalent: (i) \(D'\) can be obtained from D by a sequence of elementary operations. (ii) For any filter F in L and any filter \(F'\) in Q the number of edges in D between F and \(F'\) is not greater than the number of edges in \(D'\) between F and \(F'\).