lattice interval
Definition. Let $L$ be a lattice^{}. A subset $I$ of $L$ is called a lattice interval, or simply an if there exist elements $a,b\in L$ such that
$$I=\{t\in L\mid a\le t\le b\}:=[a,b].$$ 
The elements $a,b$ are called the endpoints of $I$. Clearly $a,b\in I$. Also, the endpoints of a lattice interval are unique: if $[a,b]=[c,d]$, then $a=c$ and $b=d$.
Remarks.

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It is easy to see that the name is derived from that of an interval on a number line. From this analogy^{}, one can easily define lattice intervals without one or both endpoints. Whereas an interval on a number line is linearly ordered^{}, a lattice interval in general is not. Nevertheless, a lattice interval $I$ of a lattice $L$ is a sublattice of $L$.

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A bounded lattice^{} is itself a lattice interval: $[0,1]$.

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A prime interval is a lattice interval that contains its endpoints and nothing else. In other words, if $[a,b]$ is prime, then any $c\in [a,b]$ implies that either $c=a$ or $c=b$. Simply put, $b$ covers $a$. If a lattice $L$ contains $0$, then for any $a\in L$, $[0,a]$ is a prime interval iff $a$ is an atom.

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Since no operations^{} of meet and join are used, all of the above discussion can be generalized to define an interval in a poset.

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Given a lattice $L$, let $\mathcal{B}$ be the collection^{} of all lattice intervals without endpoints, we can form a topolgy on $L$ with $\mathcal{B}$ as the subbasis. This does not insure that $\wedge $ and $\vee $ are continuous^{}, so that $L$ with this topological structure may not be a topological lattice.

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Locally Finite Lattice. A lattice that is derived based on the concept of lattice interval is that of a locally finite lattice. A lattice $L$ is locally finite^{} iff every one of its interval is finite. Unless the lattice is finite, a locally finite lattice, if infinite^{}, is either topless or bottomless.
Title  lattice interval 

Canonical name  LatticeInterval 
Date of creation  20130322 15:44:56 
Last modified on  20130322 15:44:56 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B99 
Classification  msc 06A06 
Defines  prime interval 
Defines  poset interval 
Defines  locally finite lattice 