# List of order theory topics

**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

## 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)*

### Orders with further algebraic operations

## 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

## Orders in mathematical logic

## 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

This article is issued from
Wikipedia.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.