Preferences and Operators

Cato, Susumu

doi:10.1007/978-981-10-1896-1_2

Susumu Cato²

Part of the book series: SpringerBriefs in Economics ((BRIEFSDBJRS))

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Abstract

This chapter focuses on how operations over binary relations work. We provide a series of basic observations on operations over binary relations and extend the classical results of Graham et al. (Complements and transitive closures. Discret Math 2(1):17–29, 1972) and Fishburn (Operations on binary relations . Discret Math 21(1):7–22, 1978). Moreover, we introduce various closure operators and clarify their implications.

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Notes

1.
The composition is also simple.
2.
(i)–(iv) of Lemma 2.2 are found in Lemma 2.1 of Fishburn (1978). He does not provide proofs.
3.
(i)–(iii) of Lemma 2.5 are found in Lemma 2.3 of Fishburn (1978). He does not provide proofs.
4.
See Berge (1963 ) for a detailed discussion of closure operators.
5.
See Bossert and Suzumura (2010) for discussions on tc and kc.
6.
Lemma 2.12 (i) is found in Lemma 2.1 of Fishburn (1978).
7.
Lemma 2.17 (i) is found in Lemma 2.3 of Fishburn (1978).
8.
The first claim of Lemma 2.18 is provided by Graham et al. (1972), and the second claim of Lemma 2.18 is provided by Cato (2012).
9.
The first claim of Lemma 2.18 is provided by Graham et al. (1972), and the second claim of Lemma 2.18 is provided by Cato (2012).

References

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Graham, R. L., Knuth, D. E., & Motzkin, T. S. (1972). Complements and transitive closures. Discrete Mathematics, 2(1), 17–29.
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Institute of Social Science, The University of Tokyo, Hongō, Bunkyō, Tokyo, Japan
Susumu Cato

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Susumu Cato
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Correspondence to Susumu Cato .

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Cato, S. (2016). Preferences and Operators. In: Rationality and Operators. SpringerBriefs in Economics(). Springer, Singapore. https://doi.org/10.1007/978-981-10-1896-1_2

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DOI: https://doi.org/10.1007/978-981-10-1896-1_2
Published: 17 August 2016
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-1895-4
Online ISBN: 978-981-10-1896-1
eBook Packages: Economics and FinanceEconomics and Finance (R0)

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