Alphabeta MathHow 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 in which every pair has a greatest lower bound, its meet , and a least upper bound, its join . A lattice is distributive when, for all ,
and
Let be a poset. An order ideal, or down-set, is a subset such that and imply . The set of all order ideals of , ordered by inclusion, is denoted . Both and are order ideals.
A lattice isomorphism is a bijection preserving meets and joins. Such a map also preserves and reflects the order, since is equivalent to .
Depends on
Used by
- A monoidal category need not be closed Counterexample
- A partition into intervals with non-monotone endpoints need not be a lattice congruence Counterexample
- A subobject lattice of an abelian category need not be distributive Counterexample
- The diamond M₃ and pentagon N₅ violate distributivity by explicit joins and meets Counterexample
- Finite lattice congruences, interval endpoints and descending rooted-chain labels Definition
- Join-irreducible elements of a nonempty finite lattice Definition
- Modular lattice Definition
- Infinite dihedral type: lower intervals are chains, but the two atoms have no upper bound Example
- The interval criterion checked on a three-element chain and a diamond Example
- The right weak interval below the longest element of A₂ is not distributive Example
- A finite lattice has a bottom and a top, and every element is the join of the join-irreducible elements below it Lemma
- Every join-irreducible element of a distributive lattice is join-prime Lemma
- Lattice quotient descent, class intervals and monotone endpoints Lemma
- Linear extensions of a finite poset: existence, prescribed initial ideals, and adjacent-swap connectivity Lemma
- The order ideals of a finite poset form a distributive lattice under union and intersection Lemma
- A poset with finite meets is a strict monoidal category Theorem
- Finite noncrossing intervals are lattices, independently of the Coxeter element Theorem
- The interval criterion for a lattice congruence: interval classes with monotone endpoints Theorem
- The right weak order interval below a fully commutative element is the lattice of order ideals of its heap Theorem
- Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics Theorem
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)