order-preserving map OrderPreservingMap 2008-01-16 11:52:34 2009-10-05 21:52:25 Definition monotonicity partial order % 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 \emph{Order-preserving map} from a poset $L$ to a poset $M$ is a function $f$ such that $$\forall x,y\in L:(x\ge y\implies f(x)\ge f(y)).$$ Order-preserving maps are also called \emph{monotone functions} or \emph{monotonic functions} or \emph{order homomorphisms} or \emph{isotone functions} or \emph{isotonic functions}. \emph{Order-reversing map} from a poset $L$ to a poset $M$ is a function $f$ such that $$\forall x,y\in L:(x\ge y\implies f(x)\le f(y)).$$ Order-reversing maps are also called \emph{antitone functions}.