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.

Join-irreducible elements of a nonempty finite lattice

Definition

Let $L$ be a nonempty finite lattice. Its least element, whose existence is proved in A finite lattice has a bottom and a top, and every element is the join of the join-irreducible elements below it ↗, is denoted $0_{L}$ . An element $j \in L$ is join-irreducible if $j \neq 0_{L}$ and

$j = a \lor b ⟹ j = a or j = b$

for all $a, b \in L$ . The set of join-irreducible elements, with the order inherited from $L$ , is denoted $JI (L)$ .

An element $j$ is join-prime if

$j \leq a \lor b ⟹ j \leq a or j \leq b .$

Join-prime implies join-irreducible whenever $j \neq 0_{L}$ . In a distributive lattice the converse holds for join-irreducible elements.

Depends on

Lattices, distributive lattices, and order ideals

Used by

Dependency tree · two levels

2 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)