List of order structures in mathematics

In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include:

Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list.
Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities)
Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions
Total orders, orderings that specify, for every two distinct elements, which one is less than the other
Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities)
Well-orders, total orders in which every non-empty subset has a least element
Well-quasi-orderings, a class of preorders generalizing the well-orders

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

See also