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Lattice Representations of Order Ideals of Posets July 17, 2009

Posted by Martin Camacho in combinatorics.
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In this post I’ll talk about the representation of order ideals of posets as distributive lattices. You can get a very good description of posets from Stanley’s Enumerative Combinatorics, Vol 1.

First, a couple definitions:

Definition 1 An order ideal of a poset {P} is a set of elements {I} such that if {x\in I} and {y\le x} then {y\in I}


Thus ideals can be finitely generated by selecting a set of generators from {P}. We can represent these order ideals visually by drawing lattices.

Definition 2 A chain decomposition of a poset {P} is a partition of the elements of {P} into sets {C_i} such that the subposet {C_i\subset P} is identical to a chain of size {|C_i|}.

These chain decomopositions actually end up in a 1-1 correspondence with distributive lattices of posets:

Definition 3 A distributive lattice representation of a chain decomposition {\mathbf{C}=(C_1,C_2,\cdots, C_k)} of some poset {P} is a {k}-dimensional lattice {L\subset \mathbb{Z}^k} which satisfies the property that {(x_1,x_2,\cdots,x_k)\in L} iff there is an order ideal {I} of {P} such that {|I\cup C_i|=x_i}.

It turns out that the number of paths in these lattices from {\widehat{0}} to {(|C_1|,|C_2|,\cdots,|C_k|)} in {L} is equal to the number of linear extensions of {P}. This is an extremely useful tool for computing the number of linear extensions of a complex poset.

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Comments»

1. Akhil Mathew - July 17, 2009

“Thus ideals can be finitely generated by selecting a set of generators from {P}. We can represent these order ideals visually by drawing lattices.”

Are the generators here to be the maximal elements of the poset ideal?

2. Martin Camacho - July 17, 2009

In this case, yes, because after picking an initial set of elements we can only include smaller ones in our ideal.

3. Akhil Mathew - July 17, 2009

So you need to assume P finite, no? Otherwise one could consider, say, \mathbb{N}, and have the ideal be the entire space.

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