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Obtaining certain minimal elements for lattices

asked 2020-10-09 16:41:50 +0200

klaaa gravatar image

updated 2020-10-10 14:02:17 +0200

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 $S_L$.

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answered 2020-10-10 12:40:08 +0200

FrédéricC gravatar image

updated 2020-10-10 12:42:13 +0200

Like this maybe

sage: L = posets.TamariLattice(2)
sage: U = L.dual() * L
sage: S = U.subposet((x,y) for x,y in U if not L.le(y,x))
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Thank you very much. My guess for general lattices was wrong, so I deleted it.

klaaa gravatar imageklaaa ( 2020-10-10 14:01:56 +0200 )edit

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Asked: 2020-10-09 16:41:50 +0200

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Last updated: Oct 10 '20