LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $\emptyset$ , 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 ( $∣ P (A) ∣ = 2^{∣ A ∣}$ for finite $A$ ), and every subset of a finite set is finite (A subset of a finite set is finite, with $∣ B ∣ \leq ∣ A ∣$ , 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 $x \leq y$ , 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 $\emptyset$ 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 · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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