# Definition:Complete Lattice

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*This page is about Complete in the context of Lattice Theory. For other uses, see Complete.*

## Definition

### Definition 1

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

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

### 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.

## Notation

The **greatest element** and **smallest element** of a **complete lattice** are denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\top$ and $\bot$ respectively.

Some sources use $1$ for the **greatest element** and $0$ for the **smallest element**.

## Also known as

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

## Also see

- Results about
**complete lattices**can be found**here**.