- List of order theory topics
-
Order theory is a branch of mathematics that studies various kinds of 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.
Contents
Overview
- Partially ordered set
- Preorder
- Totally ordered set
- Total preorder
- Chain
- Trichotomy
- Extended real number line
- Antichain
- Strict 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
Special types of partial orders
- 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)
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 (structure)
- Complete Boolean algebra
- Orthocomplemented lattice
- 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
Main article: 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
- Canonical order
- Prewellordering
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
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