<![CDATA[WUJ]]><![CDATA[info@ejournals.eu]]>http://www.ejournals.eu/20200409ejournals.eu.article.1228803010610.4467/20842589RM.18.005.883820<![CDATA[Extension theorems; Kuratowski-Zornlemma; transfinite methods]]>00ED031001Number10102<![CDATA[Reports on Mathematical Logic]]>03<![CDATA[2018]]>01<![CDATA[Number 53]]>1A01<![CDATA[Peter Schuster]]>2A01<![CDATA[Daniel Wessel]]>0101<![CDATA[A General Extension Theorem for Directed-Complete Partial Orders]]>01en001703020PLN03<![CDATA[The typical indirect proof of an abstract extension theorem, by the Kuratowski-Zorn lemma, is based on a onestep extension argument. While Bell has observed this in case of the axiom of choice, for subfunctions of a given relation, we now consider such extension patterns on arbitrary directed-complete partial orders. By postulating the existence of so-called total elements rather than maximal ones, we can single out an immediate consequence of the Kuratowski-Zorn lemma from which quite a few abstract extension theorems can be deduced more directly, apart from certain definitions by cases. Applications include Baer’s criterion for a module to be injective. Last but not least, our general extension theorem is equivalent to a suitable form of the Kuratowski-Zorn lemma over constructive set theory.
Received 27 June 2017
AMS subject classification: 03E25, 03F65]]>012018090615000401http://dx.doi.org/10.4467/20842589RM.18.005.8838