Equivalence of Complete Semilattice and Complete Lattice

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Theorem

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


The following statements are equivalent::

$(\text 1) \quad \struct {S, \preceq}$ is a complete join semilattice
$(\text 2) \quad \struct {S, \preceq}$ is a complete meet semilattice
$(\text 3) \quad \struct {S, \preceq}$ is a complete lattice


Proof

Statement $(\text 1)$ implies Statement $(\text 2)$

Let $\struct {S, \preceq}$ be a complete join semilattice.


Let $A \subseteq S$.

Let $T = \leftset{s \in S : s}$ is a lower bound for $\rightset{A}$


By definition of complete join semilattice:

$\sup T$ exists in $\struct {S, \preceq}$

Let $a = \sup T$.


We have:

$\forall s \in A : s$ is an upper bound for $T$

By definition of supremum:

$\forall s \in A: a \preceq s$

By definition of lower bound:

$a$ is a lower bound for $A$


Since $a$ is the supremum of all lower bounds of $A$:

$a = \inf A$


Since $A$ was arbitrary:

$\forall A \subseteq S : \inf A \in S$

Hence $\struct {S, \preceq}$ is a complete meet semilattice by definition.

$\Box$

Statement $(\text 2)$ implies Statement $(\text 3)$

We have:

Statement $(\text 2)$ implies Statement $(\text 1)$

is the dual statement of:

Statement $(\text 1)$ implies Statement $(\text 2)$

by Dual Pairs (Order Theory).


We have:

Statement $(\text 1)$ implies Statement $(\text 2)$

So:

Statement $(\text 2)$ implies Statement $(\text 1)$

follows from the Duality Principle.


The result now follows directly from:

$\Box$

Statement $(\text 3)$ implies Statement $(\text 1)$

The result follows immediately from:

$\blacksquare$

Sources

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