Down-set (Order Ideal)

Subset Y of poset P closed downward under the order; dually related to up-sets.

Neighborhood — ranked by edge-count

claim

Down-sets are more convenient than up-sets for many purposes in order theory.
cites
Observation that ↓ operator is easier to work with than ↑ due to order-preservation rather than order-reversal.

concept

Partially Ordered Set (Poset)
associated_with
Fundamental structure: a set with a reflexive, antisymmetric, transitive relation.
Up-set (Principal Filter)
associated_with
Subset Y of poset P such that if x ∈ Y and x ≤ y then y ∈ Y; ordered family U(P) forms a poset.

cosine ≥ 0.65 · no typed edge

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

The order relation ≤ on a poset P is determined entirely by the down-sets in P.claim0.731
Order (versus Disorder)concept0.703
Low entropy, organized state; life maintains order by consuming negative entropy.
Dropping downmethod0.690
Dynamic condition: a menu-like reveal of subordinate content.
Middle-Range Orderframework0.684
Coherent entities at the scale of rooms and bays — roughly halfway between building volume and smallest elements — whose existence and beautiful pattern of arrangement is essential for profound building order; first noted by Ingrid King
Order provides terminology, notation, and models across diverse areas of computer science including comparisons of size, information content, and definedness.claim0.683
nondeterminismconcept0.681
Inherent in Linda because an in statement chooses one matching tuple arbitrarily; essential for many parallel patterns.
Bottom-up Selectionconcept0.678
Typical Setconcept0.678
Information-theoretic concept used to formalize why representative distributions are diverse; sequences in the typical set exhibit empirical statistics consistent with the pretraining distribution