Extension of the Zorn lemma to general nontransitive binary relations
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Let ≻ be an irreflexive (strict) binary relation on a nonempty setX. Denote the completion of ≻ by ≧, i.e.,y≧x ifx≻y does not hold. An elementx * ∈X is said to be a maximal element of ≻ onX ifx * ≧x, ∀x∈X. In this paper, an extension of the Zorn lemma to general nontrasitive binary relations (may lack antisymmetry) is established and is applied to prove existence of maximal elements for general nontrasitive (reflexive or irreflexive) binary relations on nonempty sets without assuming any topological conditions or linear structures. A necessary and sufficient condition has been also established to completely characterize the existence of maximal elements for general irreflexive nontrasitive binary relations. This is the first such result available in the literature to the best of our knowledge. Many recent known existence sults in the literature for vector optimization are shown to be special cases of our result.
Key WordsIrreflexive binary relations nontrasitive binary relations maximal elements Zorn's lemma chain dominant property convex cone preference
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