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Lexicographic preference representation: Intrinsic length of linear orders on infinite sets

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  • Knoblauch, Vicki

Abstract

The intrinsic length of a linear order is the minimum of all ordinals δ such that there is a binary-criteria lexicographic representation of the linear order in {0,1}δ. Assuming the Generalized Continuum Hypothesis, we show that, for each ordinal γ and infinite set X with cardinality κ, there exist a linear order on X such that γ is the intrinsic length of that linear order if and only if logκ≤γ≤κ. This intrinsic-length partition imposes a structure on the profusion of linear orders on an infinite set.

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  • Knoblauch, Vicki, 2023. "Lexicographic preference representation: Intrinsic length of linear orders on infinite sets," Journal of Mathematical Economics, Elsevier, vol. 105(C).
    • Handle: RePEc:eee:mateco:v:105:y:2023:i:c:s0304406823000162
    DOI: 10.1016/j.jmateco.2023.102823
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    References listed on IDEAS

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    1. Mandler, Michael & Manzini, Paola & Mariotti, Marco, 2012. "A million answers to twenty questions: Choosing by checklist," Journal of Economic Theory, Elsevier, vol. 147(1), pages 71-92.
    2. Knoblauch, Vicki, 2022. "The lexicographic complexity of asymmetric binary relations," Mathematical Social Sciences, Elsevier, vol. 117(C), pages 6-12.
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    5. Ram Sewak Dubey & Giorgio Laguzzi, 2022. "How rare are the properties of binary relations?," Papers 2202.05229, arXiv.org.
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    8. Vicki Knoblauch, 2015. "Two preference metrics provide settings for the study of properties of binary relations," Theory and Decision, Springer, vol. 79(4), pages 615-625, December.
    9. Beardon, Alan F. & Candeal, Juan C. & Herden, Gerhard & Indurain, Esteban & Mehta, Ghanshyam B., 2002. "The non-existence of a utility function and the structure of non-representable preference relations," Journal of Mathematical Economics, Elsevier, vol. 37(1), pages 17-38, February.
    10. Knoblauch, Vicki, 2000. "Lexicographic orders and preference representation," Journal of Mathematical Economics, Elsevier, vol. 34(2), pages 255-267, October.
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