---
tag: CZPBQY67ZJ
kind: concept
name: Conditionally complete lattice
msc: [06B23]
specializes: [WH4SFAACAS]
sources:
  - url: https://en.wikipedia.org/wiki/Completeness_(order_theory)
---

## Summary

A conditionally complete lattice is a lattice in which every nonempty bounded-above subset has a supremum and every nonempty bounded-below subset has an infimum.
It is the order-theoretic form of the completeness of the [[real numbers|08J2M1JYAG]].
Unlike a [[complete lattice|D7A0ZGZT5S]], it need not have a least or a greatest element.

## Explanations

### L1 human

The real numbers are not a complete lattice, because the set of all integers has no upper bound.
But every nonempty set of reals that does have an upper bound has a least one.
A conditionally complete lattice is an ordered set with this weaker property, together with the matching property for lower bounds.
The integers are another example: a nonempty set of integers that is bounded above has a largest element, which is its supremum.
The rational numbers are not an example, since the rationals below $\sqrt{2}$ have no least rational upper bound.