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Fixed‐points of Set‐continuous Operators
Author(s) -
Dzierzgowski Daniel,
Esser Olivier,
Hinnion Roland
Publication year - 2000
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/(sici)1521-3870(200005)46:2<183::aid-malq183>3.0.co;2-2
Subject(s) - mathematics , intersection (aeronautics) , axiom of choice , axiom , fixed point , operator (biology) , monotone polygon , set (abstract data type) , set theory , discrete mathematics , fixed point theorem , least fixed point , mathematical economics , pure mathematics , computer science , schauder fixed point theorem , brouwer fixed point theorem , mathematical analysis , biochemistry , chemistry , geometry , repressor , transcription factor , engineering , gene , programming language , aerospace engineering
In this paper, we study when a set‐continuous operator has a fixed‐point that is the intersection of a directed family. The framework of our study is the Kelley‐Morse theory KMC – and the Gödel‐Bernays theory GBC – , both theories including an Axiom of Choice and excluding the Axiom of Foundation. On the one hand, we prove a result concerning monotone operators in KMC – that cannot be proved in GBC – . On the other hand, we study conditions on directed superclasses in GBC – in order that their intersection is a fixed‐point of a set‐continuous operator. Finally, we illustrate our results with a solution to the liar paradox and a construction of maximal bisimulations.