LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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The order ideals of a finite poset form a distributive lattice under union and intersection

Statement

For a finite poset $P$ , the order ideals $J(P)$ form a finite distributive lattice under inclusion. Its meet is intersection, its join is union, its bottom is $\varnothing$ , and its top is $P$ .

Facts & Assumptions

Given: A finite poset $P$ and order ideals $I,J,K\in J(P)$ .

[F1]

An order ideal is downward closed, and a distributive lattice satisfies the two distributive identities for meet and join (Lattices, distributive lattices, and order ideals).

[L1]

The power set of a finite set is finite ( $\lvert\mathcal{P}(A)\rvert = 2^{\lvert A\rvert}$ for finite $A$ ), and every subset of a finite set is finite (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$ , and equality holds if and only if $B = A$ ).

Proof

technique · direct

1.1

The sets $I\cap J$ and $I\cup J$ are order ideals: if $y$ belongs to the intersection or union and $x\le y$ , downward closure in the relevant ideal puts $x$ in the same intersection or union.

givenF1

1.2

Set union and intersection satisfy $I\cap(J\cup K)=(I\cap J)\cup(I\cap K)$ and $I\cup(J\cap K)=(I\cup J)\cap(I\cup K)$ element by element.

givenalgebra

2.1

In the inclusion order, $I\cap J$ is the greatest lower bound of $I,J$ and $I\cup J$ is their least upper bound. Also $\varnothing$ and $P$ are respectively the least and greatest order ideals.

step 1.1F1

3.1

Thus $J(P)$ is a distributive lattice with the asserted operations and bounds. It is finite because it is a subcollection of the finite power set of $P$ .

step 2.1step 1.2F1L1∎

Depends on

Used by

Birkhoff representation theorem: every finite distributive lattice is isomorphic to the lattice of order ideals of its join-irreducible poset Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

MIT OpenCourseWare 18.212, Lecture 16: Distributive lattices (standard reference, not scraped)