List of order theory topics

Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another.

An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value and mathematical optimization.

Overview[edit]

Distinguished elements of partial orders[edit]

Greatest element (maximum, top, unit), Least element (minimum, bottom, zero)
Maximal element, minimal element
Upper bound
- Least upper bound (supremum, join)
- Greatest lower bound (infimum, meet)
- Limit superior and limit inferior
Irreducible element
Prime element
Compact element

Subsets of partial orders[edit]

Cofinal and coinitial set, sometimes also called dense
Meet-dense set and join-dense set
Linked set (upwards and downwards)
Directed set (upwards and downwards)
centered and σ-centered set
Net (mathematics)
Upper set and lower set
Ideal and filter
- Ultrafilter

Special types of partial orders[edit]

Completeness (order theory)
Dense order
Distributivity (order theory)
Ascending chain condition
- Infinite descending chain
Countable chain condition, often abbreviated as ccc
Knaster's condition, sometimes denoted property (K)

Well-orders[edit]

Completeness properties[edit]

Semilattice
Lattice
(Directed) complete partial order, (d)cpo
Bounded complete
Complete lattice
- Knaster–Tarski theorem
Infinite divisibility

Orders with further algebraic operations[edit]

Orders in algebra[edit]

Functions between partial orders[edit]

Completions and free constructions[edit]

Domain theory[edit]

Orders in mathematical logic[edit]

Orders in topology[edit]

Stone duality
- Stone's representation theorem for Boolean algebras
Specialization (pre)order
Order topology of a total order (open interval topology)
Alexandrov topology
Upper topology
Scott topology
- Scott continuity
Lawson topology
Finer topology

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice