%0 Journal Article
%T A General Extension Theorem for Directed-Complete Partial Orders
%A Schuster, Peter
%A Wessel, Daniel
%J Reports on Mathematical Logic
%V 2018
%R 10.4467/20842589RM.18.005.8838
%N Number 53
%P 79-96
%K Extension theorems; Kuratowski-Zornlemma; transfinite methods
%@ 0137-2904
%D 2018
%U https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/a-general-extension-theorem-for-directed-complete-partial-orders
%X 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.