Let L be a finite lattice and Lop the opposite lattice. We can then look at the product lattice U=Lop×L and inside U the poset SL= { (r1,r2)∈Lop×L|r2≰ }. My question is whether there is an easy way to obtain the poset U for a given lattice L together with the minimal elements min(S_L) of S_L.
When L is distributive min(S_L) should be equal to the set of tuples (m,row(m)) where m is meet-irreducible and row(m) denotes the rowmotion bijection applied to m (viewing elements of L as order ideals of the poset of join-irreducible). For general lattices L one should obtain a generalisation of this, maybe a canonical bijection between meet-prime and join-prime elements.