Equivalence of Definitions of Complete Lattice

Theorem

The following definitions of the concept of Complete Lattice are equivalent:

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.

Proof

Immediately apparent by definition of supremum and infimum.

$\blacksquare$