| Summary: | In Hliněná et al., Pre-orders and orders generated by uninorms, in 15th International Conference IPMU 2014, Proceedings, Part III, Montpellier, France, 2014, pp. 307–316 the authors, inspired by Karaçal and Kesicioğlu, A t-partial order obtained from t-norms, Kybernetika. 47 (2011), 300–314, introduced a pre-order induced by uninorms. This contribution is devoted to a classification of families of uninorms by means of pre-orders (and orders) they induce. Philosophically, the paper follows the original idea of Clifford, Naturally totally ordered commutative semigroups, Am. J. Math. 76 (1954), 631–646. The present paper is an extension of the paper Hliněná and Kalina, A characterization of uninorms by means of a pre-order they induce, in Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), Atlantis Press, 2019, pp. 595–601 that was presented at EUSFLAT 2019. As a by-product, we present a t-norm that possesses a single discontinuity point.
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