concept
active
concept:partially-ordered-set-posetPartially Ordered Set (Poset)
Fundamental structure: a set with a reflexive, antisymmetric, transitive relation.
Neighborhood — ranked by edge-count
Concepts (7)
concept
- Category TheorycitesFundamental mathematical tool; poset-as-category provides simple instances of categorical notions like products and adjunctions.
- Down-set (Order Ideal)associated_withSubset Y of poset P closed downward under the order; dually related to up-sets.
- Information Orderingassociated_withOrder relations modeling approximation and information content; appears in string prefixes, interval approximations, and partial maps.
- Up-set (Principal Filter)associated_withSubset Y of poset P such that if x ∈ Y and x ≤ y then y ∈ Y; ordered family U(P) forms a poset.
- Join and Meet Operationsassociated_withBinary operations in lattices; join (∨) gives supremum, meet (∧) gives infimum.
- Structure-preserving maps between posets; foundational for category-theoretic treatment of ordered structures.
- Duality PrinciplecitesAny statement about a poset P yields a dual statement about P^∂ by interchanging ≤ and ≥; permits proof of one statement to establish its dual.
Findings (1)
finding
- Established property used to characterize down-sets and to relate elements to maximal upper bounds.
Related by similarity (8)
cosine ≥ 0.65 · no typed edgeEntities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.
- Foundational hypothesis of Domain Theory: partial order structure (D, ⊑) captures information ordering without quantification.
- Motivating question throughout: using order theory to capture information flow, approximation, and program behavior.
- Mathematical representation of precedence relations among steps: which centers must be in position before another can be formed, defining good sequences as linearizations that minimize backtracking.
- David Bohm's framework where reality unfolds from an enfolded wholeness, considered by Bohm to be essentially the same as Alexander's wholeness.