# bounded lattice

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

Title bounded lattice BoundedLattice 2013-03-22 15:02:28 2013-03-22 15:02:28 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 06B05 msc 06A06 top bottom bounded poset