User:Leigh.Samphier/Topology/Definition:Complete Lattice

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

Definition

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.

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.

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.