DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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.

Lattices, distributive lattices, and order ideals

Definition

A lattice is a poset $L$ in which every pair $x, y$ has a greatest lower bound, its meet $x \land y$ , and a least upper bound, its join $x \lor y$ . A lattice is distributive when, for all $x, y, z \in L$ ,

$x \land (y \lor z) = (x \land y) \lor (x \land z)$

and

$x \lor (y \land z) = (x \lor y) \land (x \lor z) .$

Let $P$ be a poset. An order ideal, or down-set, is a subset $I \subseteq P$ such that $y \in I$ and $x \leq y$ imply $x \in I$ . The set of all order ideals of $P$ , ordered by inclusion, is denoted $J (P)$ . Both $\emptyset$ and $P$ are order ideals.

A lattice isomorphism is a bijection preserving meets and joins. Such a map also preserves and reflects the order, since $x \leq y$ is equivalent to $x \land y = x$ .

Depends on

Partial order and partially ordered set

Used by

Dependency tree · one level

1 result within one dependency step 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)