branch Branch 2002-07-26 20:40:01 2002-07-26 20:40:01 Definition branch cofinal branch % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \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} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} A subset $B$ of a tree $(T,<_T)$ is a \emph{branch} if $B$ is a maximal linearly ordered subset of $T$. That is: \begin{itemize} \item $<_T$ is a linear ordering of $B$ \item If $t\in T\setminus B$ then $B\cup \{t\}$ is not linearly ordered by $<_T$. \end{itemize} This is the same as the intuitive conception of a branch: it is a set of nodes starting at the root and going all the way to the tip (in infinite sets the conception is more complicated, since there may not be a tip, but the idea is the same). Since branches are maximal there is no way to add an element to a branch and have it remain a branch. A \emph{cofinal branch} is a branch which intersects every level of the tree.