A Generalization of Arrow’s Lemma on Extending a Binary Relation
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 c...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2019/5397036 |
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| author | Athanasios Andrikopoulos |
| author_facet | Athanasios Andrikopoulos |
| author_sort | Athanasios Andrikopoulos |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e594caf07544413fa0d5cd3a3d8a6944 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e594caf07544413fa0d5cd3a3d8a69442025-02-03T01:22:51ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252019-01-01201910.1155/2019/53970365397036A Generalization of Arrow’s Lemma on Extending a Binary RelationAthanasios Andrikopoulos0Department of Computer Engineering & Informatics, University of Patras, GreeceBy 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.http://dx.doi.org/10.1155/2019/5397036 |
| spellingShingle | Athanasios Andrikopoulos A Generalization of Arrow’s Lemma on Extending a Binary Relation International Journal of Mathematics and Mathematical Sciences |
| title | A Generalization of Arrow’s Lemma on Extending a Binary Relation |
| title_full | A Generalization of Arrow’s Lemma on Extending a Binary Relation |
| title_fullStr | A Generalization of Arrow’s Lemma on Extending a Binary Relation |
| title_full_unstemmed | A Generalization of Arrow’s Lemma on Extending a Binary Relation |
| title_short | A Generalization of Arrow’s Lemma on Extending a Binary Relation |
| title_sort | generalization of arrow s lemma on extending a binary relation |
| url | http://dx.doi.org/10.1155/2019/5397036 |
| work_keys_str_mv | AT athanasiosandrikopoulos ageneralizationofarrowslemmaonextendingabinaryrelation AT athanasiosandrikopoulos generalizationofarrowslemmaonextendingabinaryrelation |