TY - JOUR TI - A General Extension Theorem for Directed-Complete Partial Orders AU - Schuster, Peter AU - Wessel, Daniel TI - A General Extension Theorem for Directed-Complete Partial Orders AB - 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. VL - 2018 IS - Number 53 PY - 2018 SN - 0137-2904 C1 - 2084-2589 SP - 79 EP - 96 DO - 10.4467/20842589RM.18.005.8838 UR - https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/a-general-extension-theorem-for-directed-complete-partial-orders KW - Extension theorems; Kuratowski-Zornlemma; transfinite methods