Least Reflexive Points of Relations [chapter]

Jules Desharnais, Bernhard Möller
2008 Automatic Program Development  
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
more » ... ly 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.
doi:10.1007/978-1-4020-6585-9_14 fatcat:eghnt4nbmvgqjmokaw2z3ro6cy