List Of Order Topics

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 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 partial order

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
• Relative complement
• 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
• Infinite tree (graph theory)
• Suslin's problem
• Absorption law
• Canonical order

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

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