---
tag: A5D94C38HZ
kind: definition
name: Conditionally complete lattice
msc: [06B23]
defines: CZPBQY67ZJ
uses: [WH4SFAACAS, 0S60DVMVPN, 0R3H5PANRQ, 0QF1MZ9FRG, 08J2M1JYAG, D7A0ZGZT5S]
sources:
  - url: https://en.wikipedia.org/wiki/Completeness_(order_theory)
---

## Definition

Let $(L, \le)$ be a [[lattice|WH4SFAACAS]].
It is a *conditionally complete lattice* if:

1. every nonempty subset of $L$ that has an [[upper bound|0S60DVMVPN]] in $L$ has a [[supremum|0R3H5PANRQ]] in $L$, and
2. every nonempty subset of $L$ that has a lower bound in $L$ has an [[infimum|0QF1MZ9FRG]] in $L$.

## Remarks

Clause 2 follows from clause 1.
Indeed, the infimum of a nonempty set $S$ bounded below is the supremum of the set of lower bounds of $S$.
That set is nonempty and bounded above by any element of $S$.
Example: the [[real numbers|08J2M1JYAG]] with the usual order, by the completeness axiom.
Non-example: $\Q$, since $\set{q \in \Q : q^2 < 2}$ is nonempty and bounded above by $2$ but has no supremum in $\Q$.
Every [[complete lattice|D7A0ZGZT5S]] is conditionally complete, and a conditionally complete lattice with a least and a greatest element is complete.
A totally ordered set with property 1 is said to have the least-upper-bound property.