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.

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

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$1\wedge x=x$ and $1\vee x=1$ for all $x\in L$.

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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 $1={1}^{\prime}$.

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$0$ is called the bottom of $L$ and $1$ is called the top of $L$.

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$L$ 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  20130322 15:02:28 
Last modified on  20130322 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 