Abstract
In this paper we first consider the problem of extending a fuzzy (weak) preorder on a set W to a fuzzy relation (preorder) on subsets of W, and consider different possibilities using different forms of quantification. For each of them we propose possible definitions of corresponding indistinguishability and strict preorder relations associated to the initial preorder, both on W and on its power set \(\mathscr {P}(W)\). We compare them and we study conditions under which the strict relation is transitive.
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Notes
- 1.
This is the strict order companion defined and studied in [4].
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Acknowledgements
The authors acknowledges partial support by the Spanish FEDER/MINECO project TIN2015-71799-C2-1-P.
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Dedication
This paper is our humble homage to the memory of Pedro Gil. Excellent researcher and better person, he has been one of the pioneers of fuzzy sets in Spain and founder and driving force of the research group on fuzzy sets at the University of Oviedo. Our contribution is devoted to fuzzy preorders and their decomposition, a subject that was very close to the research interests of Pedro Gil. Along many years, we jointly participated in many events and we have enjoyed his friendship and shared many unforgettable moments.
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Esteva, F., Godo, L. (2018). On Extending Fuzzy Preorders to Sets and Their Corresponding Strict Orders. In: Gil, E., Gil, E., Gil, J., Gil, M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-319-73848-2_54
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DOI: https://doi.org/10.1007/978-3-319-73848-2_54
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