Editing List of order theory topics

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[[Order theory]] is a branch of [[mathematics]] that studies various kinds of objects (often [[binary relation]]s) 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 (mathematics)|inequality]], [[extreme value]] and [[mathematical optimization]].

==Overview==
*[[Partially ordered set]]
*[[Preorder]]
*[[Totally ordered set]]
**[[Total preorder]]
**[[Chain (order theory)|Chain]]
**[[Trichotomy (mathematics)|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]] (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 (order theory)|Irreducible element]]
*[[Prime element (order theory)|Prime element]]
*[[Compact element]]

==Subsets of partial orders==

*[[Cofinal (mathematics)|Cofinal]] and [[Cofinal (mathematics)|coinitial]] set, sometimes also called [[Cofinal (mathematics)|dense]]
*[[Meet-dense set]] and [[join-dense set]]
*[[Linked set]] (upwards and downwards)
*[[Directed set]] (upwards and downwards)
*[[Centered set|centered]] and [[centered set|σ-centered set]]
*[[Net (mathematics)]]
*[[Upper set]] and lower set
*[[ideal (order theory)|Ideal]] and [[Filter (mathematics)|filter]]
**[[Ultrafilter]]

==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]]
*[[Countable chain condition]], often abbreviated as ''ccc''
*[[Knaster's condition]], sometimes denoted ''property (K)''

=== [[Well-order]]s ===
*[[Well-founded relation]]
*[[Ordinal number]]
*[[Well-quasi-ordering]]

===[[completeness (order theory)|Completeness properties]]===
* [[Semilattice]]
* [[lattice (order)|Lattice]]
* (Directed) [[complete partial order]], (d)cpo
* [[Bounded complete]]
* [[Complete lattice]]
**[[Knaster&ndash;Tarski theorem]]
* [[Infinite divisibility]]

===Orders with further [[abstract algebra|algebraic]] operations===
*[[Heyting algebra]]
**[[Relatively complemented lattice]]
*[[Complete Heyting algebra]]
**[[Pointless topology]]
*[[MV-algebra]]
*[[Ockham algebra]]s:
**[[Stone algebra]]
**[[De Morgan algebra]]
***[[Kleene algebra (with involution)]]
***[[Łukasiewicz–Moisil algebra]]
**[[Boolean algebra (structure)]]
***[[Boolean ring]]
***[[Complete Boolean algebra]]
*[[Orthocomplemented lattice]]
*[[Quantale]]

===Orders in [[algebra]]===
*[[Partially ordered monoid]]
*[[Ordered group]]
**[[Archimedean property]]
*[[Ordered ring]]
*[[Ordered field]]
*[[Artinian ring]]
*[[Noetherian]]
*[[Linearly ordered group]]
*[[Monomial order]]
*[[Weak order of permutations]]
*[[Bruhat order]] on a Coxeter group
*[[Incidence algebra]]

==[[Function (mathematics)|Functions]] between partial orders==
*[[Monotonic]]
*[[Pointwise order]] of functions
*[[Galois connection]]
*[[Order embedding]]
*[[Order isomorphism]]
*[[Closure operator]]
*Functions that preserve suprema/infima

==[[Completion (order theory)|Completions]] and [[free object|free constructions]]==
*[[Dedekind completion]]
*[[Ideal completion]]

==Domain theory==
{{Main|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]]
*[[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]]

{{Order theory}}
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