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.
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This research is supported by a grant from NSERC (Natural Sciences and Engineering Research Council of Canada).
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Desharnais, J., Möller, B. Least Reflexive Points of Relations. Higher-Order Symb Comput 18, 51–77 (2005). https://doi.org/10.1007/s10990-005-7006-5
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DOI: https://doi.org/10.1007/s10990-005-7006-5