Extension of restricted equivalence functions and similarity measures for type-2 fuzzy sets de Miguel, Laura Santiago, Regivan Wagner, Christian Garibaldi, Jonathan M. Takáč, Zdenko Roldán López de Hierro, Antonio Francisco Bustince, Humberto Explainable AI Information loss Restricted equivalence functions Similarity measures Type-2 fuzzy sets Uncertainty Inteligencia artificial explicable Pérdida de información Funciones de equivalencia restringida Medidas de similaridad Conjunto difuso Conjunto difuso de tipo 2 Incertidumbre Artículo publicado en: https://hdl.handle.net/2454/42915 Artículo publicado en: https://academica-e.unavarra.es/server/api/core/bitstreams/f7dbdd4b-d1a4-4ec3-b23e-35a0445d589e/content In this work, we generalize the notion of restricted equivalence function for type-2 fuzzy sets, leading to the notion of extended restricted equivalence functions. We also study how under suitable conditions, these new functions recover the standard axioms for restricted equivalence functions in the real setting. Extended restricted equivalence functions allow us to compare any two general type-2 fuzzy sets and to generate a similarity measure for type-2 fuzzy sets. The result of this similarity is a fuzzy set on the same referential set (i.e., domain) as the considered type-2 fuzzy set. The latter is crucial for applications such as explainable AI and decision-making, as it enables an intuitive interpretation of the similarity within the domain-specific context of the fuzzy sets. We show how this measure can be used to compare type-2 fuzzy sets with different membership functions in such a way that the uncertainty linked to type-2 fuzzy sets is not lost. This is achieved by generating a fuzzy set rather than a single numerical value. Furthermore, we also show how to obtain a numerical value for discrete referential sets. 2025-10-24T08:32:57Z 2025-10-24T08:32:57Z 2022-09 journal article L. de Miguel, R. Santiago, C. Wagner, J.M. Garibaldi, Z. Takáč, A.F. Roldán López de Hierro, H. Bustince (2022). Extension of restricted equivalence functions and similarity measures for type-2 fuzzy sets. IEEE Transactions on Fuzzy Systems, vol. 30, no. 9, pp. 4005-4016 https://hdl.handle.net/10481/107399 10.1109/TFUZZ.2021.3136349 eng embargoed access IEEE