A General Extension Theorem for Directed-Complete Partial Orders
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RIS BIB ENDNOTEA General Extension Theorem for Directed-Complete Partial Orders
Publication date: 06.09.2018
Reports on Mathematical Logic, 2018, Number 53, pp. 79-96
https://doi.org/10.4467/20842589RM.18.005.8838Authors
A General Extension Theorem for Directed-Complete Partial Orders
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.
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Information: Reports on Mathematical Logic, 2018, Number 53, pp. 79-96
Article type: Original article
Dipartimento di Informatica, Università degli Studi di Verona Strada le Grazie 15, 37134 Verona, Italy
Leeds University
Dipartimento di Informatica, Università degli Studi di Verona Strada le Grazie 15, 37134 Verona, Italy
Published at: 06.09.2018
Received at: 27.06.2017
Article status: Open
Licence: CC BY-NC-ND
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