List of order topics

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This is a list of order topics, by Wikipedia page.

An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value, optimization (mathematics), domain theory.

Contents 1 Basic concepts 2 Distinguished elements of partial orders 3 Subsets of partial orders 4 Special types of partial orders 4.1 Well-orders 4.2 Completeness properties 4.3 Orders with further algebraic operations 4.4 Orders in abstract algebra 5 Functions between partial orders 6 Completions and free constructions 7 Domain theory 8 Orders in mathematical logic 9 Orders in topology

Basic concepts

• Partially ordered set
• Preorder
• Totally ordered set
  • Total preorder
  • Chain
  • Trichotomy
  • Extended real number line
• Antichain
• Strict order

• Hasse diagram
  • Directed acyclic graph
• Duality (order theory)
• Product order

Distinguished elements of partial orders

• Greatest element (top, unit), Least element (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

• Cofinal set
• Dense set
• Meet-dense set and join-dense set
• Directed set
• Net (mathematics)
• Upper set and lower set
• Ideal and filter
  • Ultrafilter

Special types of partial orders

• Completeness (order theory)
• Distributivity (order theory)
  • modular lattice
  • distributive lattice
  • completely distributive lattice
• Ascending chain condition
  • Infinite descending chain

Well-orders

• Well-founded relation
• Ordinal number
• Well-quasi-ordering

Completeness properties

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

Orders with further algebraic operations

• Heyting algebra
  • Relatively complemented lattice
• Complete Heyting algebra
  • Pointless topology
• Boolean algebra
  • Boolean ring
  • Kleene algebra
• Complete Boolean algebra
• Orthocomplemented lattice
• Quantale

Orders in abstract algebra

• Partially ordered monoid
• Ordered group
  • Archimedean property
• Ordered ring
• Ordered field
• Artinian
• Noetherian
• Linearly ordered group
• Monomial order
• Weak order of permutations
• Incidence algebra

Functions between partial orders

• Monotonic
• Pointwise order of functions
• Galois connection
• Order embedding
• Order isomorphism
• Closure operator
• Functions that preserve suprema/infima

Completions and free constructions

• Dedekind completion
• Ideal completion

Domain theory

• Way-below relation
• Continuous poset
  • Continuous lattice
• Algebraic poset
  • Scott domain
  • Algebraic lattice
• Scott information system
• Powerdomain
• Scott topology
• Scott continuity

Orders in mathematical logic

• Lindenbaum algebra
• Zorn's lemma
  • Hausdorff maximality theorem
• Boolean prime ideal theorem
• Ultrafilter
• Ultrafilter lemma
• Tree (set theory)
• Tree (descriptive set theory)
• Suslin's problem
• Absorption law
• Canonical order
• Prewellordering

Orders in topology

• 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