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
editDistinguished elements of partial orders
edit- 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
edit- 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
Special types of partial orders
edit- Completeness (order theory)
- Dense order
- Distributivity (order theory)
- Ascending chain condition
- Countable chain condition, often abbreviated as ccc
- Knaster's condition, sometimes denoted property (K)
- Semilattice
- Lattice
- (Directed) complete partial order, (d)cpo
- Bounded complete
- Complete lattice
- Infinite divisibility
- Monotonic
- Pointwise order of functions
- Galois connection
- Order embedding
- Order isomorphism
- Closure operator
- Functions that preserve suprema/infima
Domain theory
editOrders in mathematical logic
edit- Stone duality
- Specialization (pre)order
- Order topology of a total order (open interval topology)
- Alexandrov topology
- Upper topology
- Scott topology
- Lawson topology
- Finer topology