# List of order topics

This is a**list of order topics**, by Wikipedia page.

An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value, optimization (mathematics), domain theory.

## Basic concepts

- Partially ordered set
- Preorder
- Totally ordered set
- Antichain
- Strict order
- Hasse diagram
- Duality (order theory)
- Product order

## Distinguished elements of partial orders

- Greatest element (top, unit), Least element (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 set
- Dense set
- Meet-dense and join-dense set
- Directed set
- Net (mathematics)
- Upper set and lower set
- Ideal and filter

## Special types of partial orders

- Completeness (order theory)
- Distributivity (order theory)
- modular lattice
- distributive lattice
- completely distributive lattice

- Ascending chain condition

### Well-orders

### Completeness properties

- Semilattice
- Lattice
- (Directed) complete partial order, (d)cpo
- Bounded complete
- Complete lattice
- Infinite divisibility

### Orders with further algebraic operations

- Heyting algebra
- Complete Heyting algebra
- Boolean algebra
- Complete Boolean algebra
- Quantale

### Orders in abstract algebra

- Partially ordered monoid
- Ordered group
- Ordered ring
- Ordered field
- Artinian
- Noetherian
- Linearly ordered group
- Monomial order
- Weak order of permutations
- 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
- Boolean prime ideal theorem
- Ultrafilter
- Ultrafilter lemma
- Infinite tree (graph theory)
- Suslin's problem
- Absorption law
- Canonical order

## Orders in topology

- Stone duality
- Specialization (pre)order
- Order topology of a total order (open interval topology)
- Alexandrov topology
- Upper topology
- Scott topology
- Lawson topology
- Finer topology