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Obtaining certain minimal elements for lattices

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.

Obtaining certain minimal elements for lattices

Let L be a finite lattice and L^{op} the opposite lattice. We can then look at the product lattice U=L^{op} \times L and inside U the poset S_L= { (r_1,r_2 ) \in L^{op} \times L | r_2 \nleq r_1 }. My question is whether there is an easy way to obtain the poset U S_L 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.

Obtaining certain minimal elements for lattices

Let L be a finite lattice and L^{op} the opposite lattice. We can then look at the product lattice U=L^{op} \times L and inside U the poset S_L= { (r_1,r_2 ) \in L^{op} \times L | r_2 \nleq r_1 }. My question is whether there is an easy way to obtain the poset S_L 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). Thus one obtain a canonical bijection between meet and join-irreducibles in that way.

For general lattices L one should obtain a generalisation of this, maybe a canonical bijection between meet-prime and join-prime elements.

Obtaining certain minimal elements for lattices

Let L be a finite lattice and L^{op} the opposite lattice. We can then look at the product lattice U=L^{op} \times L and inside U the poset S_L= { (r_1,r_2 ) \in L^{op} \times L | r_2 \nleq r_1 }. My question is whether there is an easy way to obtain the poset S_L 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). Thus one obtain a canonical bijection between meet and join-irreducibles in that way.

For general lattices L one should obtain a generalisation of this, maybe a canonical bijection between meet-prime and join-prime elements.