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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 $U$ 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.