ClosureMap 2009-04-19 18:17:50 2009-04-19 20:34:18 Definition closure axioms dual closure fixed point \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % 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} \usepackage{pst-plot} % define commands here \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} Let $P$ be a poset. A function $c:P \to P$ is called a \emph{closure map} if \begin{itemize} \item $c$ is order preserving, \item $1_P \le c$, \item $c$ is idempotent: $c\circ c = c$. \end{itemize} If the second condition is changed to $c\le 1_P$, then $c$ is called a \emph{dual closure map} on $P$. For example, the real function $f$ such that $f(r)$ is the least integer greater than or equal to $r$ is a closure map (see Archimedean property). The rounding function $[\cdot]$ is an example of a dual closure map. A \emph{fixed point} of a closure map $c$ on $P$ is an element $x\in P$ such that $c(x)=x$. It is evident that every image point of $c$ is a fixed point: for if $x=c(a)$ for some $a\in P$, then $c(x)=c(c(a))=c(a)=x$. In the example above, any integer is a fixed point of $f$. Every closure map can be characterized by an interesting decomposition property: $c: P\to P$ is a closure map iff there is a set $Q$ and a residuated function $f: P\to Q$ such that $c=f^+\circ f$, where $f^+$ denotes the residual of $f$. Again, in the example above, $f=g^+\circ g$, where $g: \mathbb{R}\to\mathbb{Z}$ is the function taking any real number $r$ to the largest integer smaller than $r$. $g$ is residuated, and its residual is $g^+(x)=x+1$. \textbf{Remark}. Closure maps are generalizations to closure operator on a set (see the parent entry). Indeed, any closure operator on a set $X$ takes a subset $A$ of $X$ to a subset $A^c$ of $X$ satisfying the closure axioms, where Axiom 2 corresponds to condition 2 above, and Axiom 3 says the operator is idempotent. To see that the operator is order preserving, suppose $A\subseteq B$. Then $B^c = (A\cup B)^c =A^c \cup B^c$ by Axiom 4, and hence $A^c\subseteq B^c$. Axiom 1 says that the empty set $\varnothing$ is a fixed point of the operator. However, in general, this is not the case, for $P$ may not even have a minimal element, as indicated by the above example. \begin{thebibliography}{6} \bibitem{tsb} T.S. Blyth, {\em Lattices and Ordered Algebraic Structures}, Springer, New York (2005). \end{thebibliography}