bounded lattice

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

• $0\wedge x=0$ and $0\vee x=x$ for all $x\in L$.

• $1\wedge x=x$ and $1\vee x=1$ 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 $0=0\wedge 0^{\prime }=0^{\prime }\wedge 0=0^{\prime }$. 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
Canonical name	BoundedLattice
Date of creation	2013-03-22 15:02:28
Last modified on	2013-03-22 15:02:28
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06B05
Classification	msc 06A06
Defines	top
Defines	bottom
Defines	bounded poset