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Fuzzy preorders: conditional extensions, extensions and their representations

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Abstract

The crisp literature provides characterizations of the preorders that admit a total preorder extension when some pairwise order comparisons are imposed on the extended relation. It is also known that every preorder is the intersection of a collection of total preorders. In this contribution we generalize both approaches to the fuzzy case. We appeal to a construction for deriving the strict preference and the indifference relations from a weak preference relation, that allows to obtain full characterizations in the conditional extension problem. This improves the performance of the construction via generators studied earlier.

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Notes

  1. Contrary to our notation, Georgescu calls transitive consistent to those fuzzy relations such that \(R^T\) is a compatible extension of R.

  2. This proof partially replicates the proof of Theorem 2 in Alcantud and Díaz (2015). At the risk of being reiterative, we present all the steps for the sake of completeness. We insist that the formulae used here and in Alcantud and Díaz (2015) to obtain the indifference and strict preference relations are different.

  3. This implication was studied by Dasgupta and Deb (2001) and Díaz et al. (2008, 2010) for constructions unrelated to Eq. (6).

  4. We are very grateful to an anonymous referee for pointing out this stimulating possibility for research.

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Acknowledgments

Both authors acknowledge financial support by the Spanish Ministerio de Economía y Competitividad, the first one under Project ECO2015-66797-P, the second one under Project TIN2014-59543-P.

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Authors and Affiliations

  1. Facultad de Economía y Empresa, BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, E37007, Salamanca, Spain

    J. C. R. Alcantud

  2. Department of Statistics and O. R., University of Oviedo, Calvo Sotelo s/n, E33007, Oviedo, Spain

    S. Díaz

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  1. J. C. R. Alcantud
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  2. S. Díaz
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Correspondence to S. Díaz.

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Alcantud, J.C.R., Díaz, S. Fuzzy preorders: conditional extensions, extensions and their representations. Fuzzy Optim Decis Making 15, 371–396 (2016). https://doi.org/10.1007/s10700-016-9230-3

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