List of order topics

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This is a list of order topics.

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

[edit] 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

[edit] Distinguished elements of partial orders

• 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

[edit] 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

[edit] Special types of partial orders

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

[edit] Well-orders

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

[edit] Completeness properties

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

[edit] Orders with further algebraic operations

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

[edit] 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

[edit] Functions between partial orders

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

[edit] Completions and free constructions

• Dedekind completion
• Ideal completion

[edit] Domain theory

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

[edit] 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

[edit] 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