---
tag: Q72NHN7ATC
kind: concept
name: Completeness in order theory
msc: [06B23, 06A06]
sources:
  - url: https://en.wikipedia.org/wiki/Completeness_(order_theory)
same_as:
  wikidata: Q5156540
  wikipedia: Completeness (order theory)
---

## Summary

In order theory, a completeness property of a partially ordered set says that every subset of a certain kind has a supremum, or an infimum.
All subsets give [[complete lattices|D7A0ZGZT5S]], and nonempty bounded subsets give [[conditionally complete lattices|CZPBQY67ZJ]].
Directed subsets give [[directed-complete partial orders|7TSXP95QN6]].
These notions organize much of lattice theory and domain theory.

## Explanations

### L1 human

An ordered set is "complete" when some collections of its elements are sure to have a least upper bound.
Which collections are required depends on the context.
Asking it for all finite nonempty sets gives a join-semilattice; asking it for every set gives a complete lattice.
The real numbers are complete for nonempty sets that are bounded above, and this is the form of completeness used in calculus.