Higher-Order and Symbolic Computation

, Volume 18, Issue 1–2, pp 51–77

Least Reflexive Points of Relations

Article

Abstract

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 relation-algebraic 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.

Keywords

least reflexive point greatest reflexive point fixed point lattice partial order relation inflationary relation 

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Département d’InformatiqueUniversité LavalQuébecCanada
  2. 2.Institut für InformatikUniversität AugsburgAugsburgGermany

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