DiscussionNote on “On the extension of nullnorms and uninorms to fuzzy truth values” [Fuzzy Sets Syst. 352 (2018) 92-118]
Introduction
Type-2 fuzzy sets were introduced by Zadeh in 1975 [31], as an extension of type-1 fuzzy sets, and have been widely investigated in mathematical theory (see, e.g. [9], [10], [11], [17], [18], [25], [32]) and engineering application. For example, type-2 fuzzy sets have applications in approximation [28], control [1], [5], [24], decision making [4], clustering [16], databases [19] and so on. The membership grades of type-2 fuzzy sets are functions from to , which are referred to as fuzzy truth values.
Recently, operations on type-2 fuzzy sets become a hot research topic, such as type-2 t-norms (see [7], [13], [14], [20], [26]), aggregation operations (see [21], [22], [27]), negations [12], [23] and type-2 implications [8], [28]. In particular, nullnorms [2] and uninorms [30] are aggregation operations [3] with neutral elements and absorbing elements on , respectively. They are generalizations of t-norms and t-conorms as well. Naturally, in [29] Xie extended nullnorms and uninorms to fuzzy truth values and investigated whether the extended nullnorms and extended uninorms form respectively type-2 nullnorms and type-2 uninorms on the algebra of fuzzy truth values. The work in [29] is interesting, but some of the results are incorrect.
This note is organized as follows. In Section 2, we recall some essential concepts and properties. In Section 3, we point out the incorrect results in [29] by counterexamples and provide the correct versions in turn. In addition, Proposition 3.7 in [29] is correct, but its proof contains a flaw. We point out the flaw and give a more general result than Proposition 3.7 in [29]. In the final section, our research is concluded.
Section snippets
Preliminary
In this section, we recall some essential requirements which are used in the sequel. The notations and terminologies used in this paper are as consistent as possible with those in [29].
Counterexamples and corrections
In what follows, All t-norms T and t-conorms S are required to be continuous, including underlying t-norms and underlying t-conorms of nullnorms (resp. uninorms). In this section, we show by counterexamples that Corollaries 2.1, 3.1, 4.1 and 4.2, Lemmas 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8, Propositions 3.4, 3.5, 3.10, 3.11, 4.1 and 4.4, Theorems 3.1, 3.3, 4.1 and 4.3 in [29] contain some flaws. And then we provide the correct versions. In addition, Proposition 3.7 in [29] is correct, but its proof
Conclusions
For Corollaries 2.1, 3.1, 4.1 and 4.2, Lemmas 3.3, 3.4, 3.5, 3.6, 3.7 and 3.8, Propositions 3.4, 3.5, 3.10, 3.11, 4.1 and 4.4, Theorems 3.1, 3.3, 4.1 and 4.3 in [29], this note illustrates by counterexamples that these results are incorrect and provides the correct versions in turn. In addition, Proposition 3.7 in [29] is correct, but its proof contains a flaw. We point out the flaw and give a more general result than Proposition 3.7 in [29]. To intuitively show our modifications to incorrect
Acknowledgement
The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11971365 and 11571010).
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