bounded lattice

A lattice $L$ is said to be bounded from below if there is an element $0\in L$ such that $0\leq x$ for all $x\in L$ . Dually, $L$ is bounded from above if there exists an element $1\in L$ such that $x\leq1$ for all $x\in L$ . A bounded lattice is one that is bounded both from above and below.

For example, any finite lattice $L$ is bounded, as $\bigvee L$ and $\bigwedge L$ , being join and meet of finitely many elements, exist. $\bigvee L=1$ and $\bigwedge L=0$ .

Remarks. Let $L$ be a bounded lattice with $0$ and $1$ as described above.

• $0\land x=0$ and $0\lor x=x$ for all $x\in L$ .
• $1\land x=x$ and $1\lor x=1$ for all $x\in L$ .
• As a result, $0$ and $1$ , if they exist, are necessarily unique. For if there is another such a pair $0^{\prime}$ and $1^{\prime}$ , then $0=0\land 0^{\prime}=0^{\prime}\land 0=0^{\prime}$ . Similarly $1=1^{\prime}$ .
• $0$ is called the bottom of $L$ and $1$ is called the top of $L$ .
• $L$ is a lattice interval and can be written as $[0,1]$ .

Remark. More generally, a poset $P$ is said to be bounded if it has both a greatest element $1$ and a least element $0$ .