criteria for a poset to be a complete lattice

Proposition. Let $L$ be a poset. Then the following are equivalent.

1. $L$ is a complete lattice.

2. for every subset $A$ of $L$, $\bigvee A$ exists.

3. for every finite subset $F$ of $L$ and every directed set $D$ of $L$, $\bigvee F$ and $\bigvee D$ exist.

Proof.

Implications $1.\Rightarrow 2.\Rightarrow 3.$ are clear. We will show $3.\Rightarrow 2.\Rightarrow 1.$

$(3.\Rightarrow 2.)$ If $A=\mathrm{\varnothing }$, then $\bigvee A=0$ by definition. So assume $A$ be a non-empty subset of $L$. Let $A^{\prime }$ be the set of all finite subsets of $A$ and $B=\{\bigvee F\mid F\in A^{\prime }\}$. By assumption, $B$ is well-defined and $A\subseteq B$. Next, let $B^{\prime }$ be the set of all directed subsets of $B$, and $C=\{\bigvee D\mid D\in B^{\prime }\}$. By assumption again, $C$ is well-defined and $B\subseteq C$. Now, every chain in $C$ has a maximal element in $C$ (since a chain is a directed set), $C$ itself has a maximal element $d$ by Zorn’s Lemma. We will show that $d$ is the least upper bound of elments of $A$. It is clear that each $a\in A$ is bounded above by $d$ ($A\subseteq B\subseteq C$). If $t$ is an upper bound of elements of $A$, then it is an upper bound of elements of $B$, and hence an upper bound of elements of $C$, which means $d\le t$.

$(2.\Rightarrow 1.)$ By assumption $\bigvee \mathrm{\varnothing }$ exists ($=0$), so that $\bigwedge L=0$. Now suppose $A$ is a proper subset of $L$. We want to show that $\bigwedge A$ exists. If $A=\mathrm{\varnothing }$, then $\bigwedge A=\bigvee L=1$ by definition of an arbitrary meet over the empty set. So assume $A\ne \mathrm{\varnothing }$. Let $A^{\prime }$ be the set of lower bounds of $A$: $A^{\prime }=\{x\in L\mid x\le a\text{ for all }a\in A\}$ and let $b=\bigvee A^{\prime }$, the least upper bound of $A^{\prime }$. $b$ exists by assumption. Since $A$ is a set of upper bounds of $A^{\prime }$, $b\le a$ for all $a\in A$. This means that $b$ is a lower bound of elements of $A$, or $b\in A^{\prime }$. If $x$ is any lower bound of elements of $A$, then $x\le b$, since $x$ is bounded above by $b$ ($b=\bigvee A^{\prime }$). This shows that $\bigwedge A$ exists and is equal to $b$.∎

Remarks.

• Dually, a poset is a complete lattice iff every subset has an infimum iff infimum exists for every finite subset and every directed subset.

• The above proposition shows, for example, that every closure system is a complete lattice.