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 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 (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 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 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-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 MV-algebra Ockham algebras: 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
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 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