Indistinguishability operators and generalized distances: the transformation problem revisited

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Science, Universitat de les Illes Balears

2 Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politécnica de Valencia

Abstract

In this paper we characterize those functions that induce a fuzzy preorder from a quasi-pseudo-metric even when the considered t-norm is not continuous. On the one hand, we prove that they must be decreasing and fulfill a special property of dominance with respect to the ordinary addition and the t-norm T under consideration. On the other hand, we have shown that they must transform asymmetric triangular triplets into asymmetric T-triangular triplets. Moreover, we also study the case in which fuzzy preorders are exactly indistinguishability operators. Concretely, we show that the monotonicity of the function and the previously mentioned dominance are sufficient but not necessary conditions. In addition, we prove that such functions must transform triangular triplets into T-triangular triplets. The developed theory is illustrated by means of appropriate examples. Furthermore, we prove that the well-known technique based on the use of the pseudo-inverses of the additive generators of continuous and Archimedean t-norms is recovered as a particular case of the new one presented here.

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References

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Iranian Journal of Fuzzy Systems
Volume 22, Issue 5
September and October 2025
Pages 97-109

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