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
See also
- v
- t
- e
Order theory
- Binary relation
- Boolean algebra
- Cyclic order
- Lattice
- Partial order
- Preorder
- Total order
- Weak ordering
- Antisymmetric
- Asymmetric
- Boolean algebra
- Completeness
- Connected
- Covering
- Dense
- Directed
- (Partial) Equivalence
- Foundational
- Heyting algebra
- Homogeneous
- Idempotent
- Lattice
- Reflexive
- Partial order
- Prefix order
- Preorder
- Semilattice
- Semiorder
- Symmetric
- Total
- Tolerance
- Transitive
- Well-founded
- Well-quasi-ordering (Better)
- (Pre) Well-order
- Alexandrov topology & Specialization preorder
- Ordered topological vector space
- Normal cone
- Order topology
- Order topology
- Topological vector lattice
- Antichain
- Cofinal
- Cofinality
- Comparability
- Duality
- Filter
- Hasse diagram
- Ideal
- Net
- Subnet
- Order morphism
- Order type
- Ordered field
- Ordered vector space
- Partially ordered group
- Upper set
- Young's lattice