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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|r2r1 }. 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(SL) of SL.

When L is distributive min(SL) 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 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|r2r1 }. My question is whether there is an easy way to obtain the poset U SL for a given lattice L together with the minimal elements min(SL) of SL.

When L is distributive min(SL) 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 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|r2r1 }. My question is whether there is an easy way to obtain the poset SL for a given lattice L together with the minimal elements min(SL) of SL.

When L is distributive min(SL) 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 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|r2r1 }. My question is whether there is an easy way to obtain the poset SL for a given lattice L together with the minimal elements min(SL) of SL.

When L is distributive min(SL) 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.