Pacific Journal of Mathematics Vol. 82, No. 1, 1979

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src="https://eye.msp.org/image.gif" alt="" /></noscript> </td> <td class="content-column" id="content-area"> <table cellspacing="0" cellpadding="0"> <tr> <td class="title-area"> <a class="title" href= "/pjm/1979/82-1/pjm-v82-n1-p04-s.pdf">Constructive versions of Tarski’s fixed point theorems</a> <h3>Patrick Cousot and Radhia Cousot</h3> </td> </tr> <tr> <td> <div class="page-numbers"> Vol. 82 (1979), No. 1, 43–57 </div> </td> </tr> <tr> <td> <div class="paper-doi"> DOI: <a href="http://dx.doi.org/10.2140/pjm.1979.82.43">10.2140/pjm.1979.82.43</a> </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Abstract</h5> </td> </tr> <tr> <td class="article-area"> <p class = "indent">Let <span class="cmmi-10">F </span>be a monotone operator on the complete lattice <span class="cmmi-10">L </span>into itself. Tarski’s lattice theoretical fixed point theorem states that the set of fixed points of <span class="cmmi-10">F </span>is a nonempty complete lattice for the ordering of <span class="cmmi-10">L</span>. We give a constructive proof of this theorem showing that the set of fixed points of <span class="cmmi-10">F </span>is the image of <span class="cmmi-10">L </span>by a lower and an upper preclosure operator. These preclosure operators are the composition of lower and upper closure operators which are defined by means of limits of stationary transfinite iteration sequences for <span class="cmmi-10">F</span>. In the same way we give a constructive characterization of the set of common fixed points of a family of commuting operators. Finally we examine some consequences of additional semi-continuity hypotheses.</p> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Mathematical Subject Classification 2000</h5> <div class="keywords"> Primary: 06A23 </div> <div class="keywords"> Secondary: 03D20 </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Milestones</h5> <div class="keywords"> Received: 5 December 1977 </div> <div class="keywords"> Revised: 26 June 1978 </div> <div class="keywords"> Published: 1 May 1979 </div> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="article"> <tr> <td class="article-area"> <h5>Authors</h5> </td> </tr> </table> <table cellspacing="0" cellpadding="0" class="author-contact"> <tr> <td class="author-name" colspan="2">Patrick Cousot</td> </tr> <tr> <td></td> </tr> <tr> <td></td> </tr> <tr> <td class="author-name" colspan="2">Radhia Cousot</td> </tr> <tr> <td></td> </tr> <tr> <td></td> </tr> </table> </td> </tr> <tr> <td> <table cellspacing="0" cellpadding="0" class="pause"> <tr> <td class="action-end"><img src="/pjm/etc/z.gif" alt="" /></td> </tr> </table> </td> <td id="content-area-end"></td> </tr> </table> <table cellspacing="0" cellpadding="0" id="footer-area"> <tr> <td class="box-footer" onclick= "javascript:window.location='http://mathscipub.org/';"> <img src="/mod/images/msp-logo.png" alt= "Mathematical Sciences Publishers" /> <p class="copyright">© Copyright 1979 Pacific Journal of Mathematics. 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Pacific Journal of Mathematics Vol. 82, No. 1, 1979