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Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets

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  • Tian, Guoqiang
  • Zhou, Jianxin

Abstract

This paper systematically studies the existence of maximal elements for unordered binary relation on compact or noncompact sets in a general topological space. This is done by developing a method, called transfer method, to derive various necessary and sufficient conditions that characterize the existence of maximal elements for a binary relation in terms of:(1) (generalized) transitivity conditions under certain topological assumptions;(2) topological conditions under certain (generalized) transitivity assumptions; and (3) (generalized)convexity conditions under certain topological assumptions. There are two basic approaches in the literature to prove the existence by providing sufficient conditions. One assumes certain convexity and continuity conditions for a topological vector space and the other assumes certain weakened transitivity and continuity conditions for a general topological space. The results unify those two approaches and generalize almost all of the existing results in literature.

Suggested Citation

  • Tian, Guoqiang & Zhou, Jianxin, 1992. "Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets," MPRA Paper 41227, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:41227
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    File URL: https://mpra.ub.uni-muenchen.de/41227/1/MPRA_paper_41227.pdf
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    References listed on IDEAS

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    1. Jonathan M. Borwein, 1983. "On the Existence of Pareto Efficient Points," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 64-73, February.
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    Cited by:

    1. Tian, Guoqiang, 2015. "On the existence of equilibria in games with arbitrary strategy spaces and preferences," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 9-16.
    2. M. Carmen Sánchez & Juan-Vicente Llinares & Begoña Subiza, 2003. "A KKM-result and an application for binary and non-binary choice functions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(1), pages 185-193, January.
    3. Leandro Nascimento, 2011. "Remarks on the consumer problem under incomplete preferences," Theory and Decision, Springer, vol. 70(1), pages 95-110, January.
    4. Tian, Guoqiang, 2012. "A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem," MPRA Paper 57929, University Library of Munich, Germany, revised Jul 2014.
    5. Quartieri, Federico, 2021. "Existence of maximals via right traces," MPRA Paper 107189, University Library of Munich, Germany.
    6. Didier Aussel & Parin Chaipunya, 2024. "Variational and Quasi-Variational Inequalities Under Local Reproducibility: Solution Concept and Applications," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1531-1563, November.
    7. M. Ali Khan & Metin Uyanik, 2021. "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 799-840, April.
    8. Nosratabadi, Hassan, 2014. "Partially upper continuous preferences: Representation and maximal elements," Economics Letters, Elsevier, vol. 125(3), pages 408-410.
    9. Rodriguez-Palmero, Carlos & Garcia-Lapresta, Jose-Luis, 2002. "Maximal elements for irreflexive binary relations on compact sets," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 55-60, January.
    10. Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
    11. Guoqiang Tian, 2016. "On the existence of price equilibrium in economies with excess demand functions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 5-16, April.

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    More about this item

    Keywords

    Binary relations; maximal elements; transfer continuities; transfer transitivities; transfer convexities;
    All these keywords.

    JEL classification:

    • D50 - Microeconomics - - General Equilibrium and Disequilibrium - - - General

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