# Definition:Complete Lattice

## Definition

### Definition 1

Let $\struct {S, \preceq}$ be a lattice.

Then $\struct {S, \preceq}$ is a complete lattice if and only if:

$\forall T \subseteq S: T$ admits both a supremum and an infimum.

### Definition 2

Let $\struct {S, \preceq}$ be an ordered set.

Then $\struct {S, \preceq}$ is a complete lattice if and only if:

$\forall S' \subseteq S: \inf S', \sup S' \in S$

That is, if and only if all subsets of $S$ have both a supremum and an infimum.

## Also known as

A complete lattice is also known as a complete ordered set.