LatticeInterval 2006-03-08 12:52:49 2007-05-23 08:29:56 Definition prime interval poset interval locally finite lattice \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % define commands here \textbf{Definition}. Let $L$ be a lattice. A subset $I$ of $L$ is called a \emph{lattice interval}, or simply an \emph{\PMlinkescapetext{interval}} if there exist elements $a,b\in L$ such that $$I=\lbrace t\in L\mid a\le t\le b\rbrace:=[a,b].$$ 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 $b=d$. \textbf{Remarks}. \begin{itemize} \item 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 $L$. \item A bounded lattice is itself a lattice interval: $[0,1]$. \item A \emph{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. \item Since no operations of meet and join are used, all of the above discussion can be generalized to define an interval in a poset. \item 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. \item \textbf{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. \end{itemize}