Least reflexive points of relations

  • Assume a partially ordered set (S,<=) and a relation R on S. We consider various sets of conditions in order to determine whether they ensure the existence of a least reflexive point, that is, a least x such that xRx. This is a generalization of the problem of determining the least fixed point of a function and the conditions under which it exists. To motivate the investigation we first present a theorem by Cai and Paige giving conditions under which iterating R from the bottom element necessarily leads to a minimal reflexive point; the proof is by a concise relationalgebraic calculation. Then, we assume a complete lattice and exhibit sufficient conditions, depending on whether R is partial or not, for the existence of a least reflexive point. Further results concern the structure of the set of all reflexive points; among other results we give a sufficient condition that these form a complete lattice, thus generalizing Tarski's classical result to the nondeterministic case.

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Metadaten
Author:Jules DesharnaisGND, Bernhard MöllerGND
URN:urn:nbn:de:bvb:384-opus4-2080
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/257
Series (Serial Number):Reports / Technische Berichte der Fakultät für Angewandte Informatik der Universität Augsburg (2002-13)
Publisher:Institut für Informatik, Universität Augsburg
Place of publication:Augsburg
Type:Report
Language:English
Year of first Publication:2002
Publishing Institution:Universität Augsburg
Release Date:2006/06/20
Tag:Least reflexive point; greatest reflexive point; fixed point; lattice; partial order; relation; inflationary relation
Institutes:Fakultät für Angewandte Informatik
Fakultät für Angewandte Informatik / Institut für Informatik
Fakultät für Angewandte Informatik / Institut für Informatik / Professur für Programmiermethodik und Multimediale Informationssysteme
Dewey Decimal Classification:0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik