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A Generalization of Arrow’s Lemma on Extending a Binary Relation

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  • Athanasios Andrikopoulos

Abstract

By examining whether the individualistic assumptions used in social choice could be used in the aggregation of individual preferences, Arrow proved a key lemma that generalizes the famous Szpilrajn’s extension theorem and used it to demonstrate the impossibility theorem. In this paper, I provide a characterization of Arrow’s result for the case in which the binary relations I extend are not necessarily transitive and are defined on abelian groups. I also give a characterization of the existence of a realizer of a binary relation defined on an abelian group. These results also generalize the well-known extension theorems of Szpilrajn, Dushnik-Miller, and Fuchs.

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  • Athanasios Andrikopoulos, 2019. "A Generalization of Arrow’s Lemma on Extending a Binary Relation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2019, pages 1-6, April.
    • Handle: RePEc:hin:jijmms:5397036
    DOI: 10.1155/2019/5397036
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    References listed on IDEAS

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    1. Herden, Gerhard & Pallack, Andreas, 2002. "On the continuous analogue of the Szpilrajn Theorem I," Mathematical Social Sciences, Elsevier, vol. 43(2), pages 115-134, March.
    2. Stephen A. Clark, 1988. "An extension theorem for rational choice functions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 55(3), pages 485-492.
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