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:

$\forall T \subseteq S: T$ 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.


Also see