Notes on type-2 triangular norms and their residual operators
Introduction
Mendel [10], [17] pointed out that there are uncertainties in fuzzy logic systems. For example, the meaning of the words used in the antecedents and consequents of values can be uncertain, that is, words can mean different things to different people. However, ordinary fuzzy sets [21] cannot reflect these uncertainties, since they are characterized by crisp membership functions. Type-2 fuzzy sets were proposed by Zadeh [22] to model and manipulate these uncertainties, whose truth values are ordinary fuzzy sets on the unit interval and called fuzzy truth values. Type-2 fuzzy sets have been applied in many areas, such as decision making [1], [25], control [4], [15], pattern recognition [2], [16] and clustering [5], [9], [16], [23].
Type-2 t-norm is a favorite topic in the study of type-2 fuzzy sets. Gera and Dombi [6] studied the exact calculations of extended t-(co)norms on fuzzy truth values, which were further discussed in [13]. Starczewski [18] investigated extended t-norms on fuzzy truth interval or fuzzy truth numbers. Hu and Kwong [7] discussed t-norm extension operations. Moreover, Wang and Hu [20] studied the lattice structure of the algebra of fuzzy values. Based on it, fuzzy-valued t-norms and their residual operations were discussed on the algebra of fuzzy values, when t-norms are left continuous.
Recently, Li [14] investigated T-extension operations of t-(co)norms and their residual operators on the algebra of fuzzy truth values, where T denotes a t-norm. In [14], Li constructed type-2 t-norms on the algebra of fuzzy truth values with respect to the partial order ⩽ and the partial order ⊑, respectively. Moreover, it was pointed out that extended minimum and its residual operation with respect to the partial order ⊑ form a BL-algebra on the algebra of convex normal and upper semicontinuous fuzzy truth values. Li also pointed out that the extended operation of a continuous Archimedean t-norm and its residual operation with respect to the partial order ⊑ form a BL-algebra on the family of decreasing, normal and upper semicontinuous fuzzy truth values. However, there are some flaws in [14]. For example, some calculations of T-extension of t-(co)norms and their residual operations are incorrect; the closedness of type-2 t-norms are unverified. Moreover, it is incorrect that extended minimum and the extension of a continuous Archimedean t-norm with their corresponding residual operations form BL-algebras. For the convenience of readers, we present the flaws mentioned above with counterexamples and provide the correct versions in our paper.
The content of the paper is organized as follows. In Section 2, we recall some fundamental concepts and related properties of t-norms and fuzzy truth values. In Section 3, we further investigate type-2 t-norms and their residual operations with respect to the ordinary partial order ⩽ and give examples to show the flaws in [14]. Section 4 discusses type-2 t-norms and their residual operations with respect to the partial order ⊑. Moreover, we study the algebraic structures of type-2 t-norms and their residual operations with respect to the partial order ⊑. In the final section, we present some conclusions of our research.
Section snippets
Preliminaries
In this section, we present some basic concepts and terminology used throughout the paper.
A fuzzy truth value f is a mapping from the unit interval [0, 1] to itself. The family of all fuzzy truth values is denoted as . The operations on fuzzy truth values are defined as follows: for all and x ∈ [0, 1], where ∧, ∨, and ′ are minimum, maximum and involutive negation on [0, 1], respectively. The order relation on fuzzy truth values is
Type-2 t-norms and its residual operations on
In this section, we show the flaws in the study of type-2 triangular norms and their residual operations with respect to the ordinary partial order ⩽ in [14] with counterexamples. Moreover, we provide the correct versions.
In [14], Li listed calculations of T-extension of a continuous t-norm * (see Theorem 3.1 in [14]). However, Eqs. (15) and (17) in [14] are incorrect. The example is given as follows.
Example 3.1 Let t-norm * be the ordinal sum of a family of continuous Archimedean t-norms as
Type-2 t-norms and its residual operators on
In this section, we study type-2 triangular norms and their residual operations on .
In [14], Li pointed out that for all t-norms *, T and t-conorm ⊗ on [0, 1], the T-extension operations *T and ⊗T are closed on (see Proposition 4.3 in [14]). However, Li applied Lemma 3.9 (Proposition 2.12 in [7]) in the proof of that conclusion. Hence the continuity of t-norms *, T and t-conorm ⊗ is required. Moreover, the conclusion mentioned above need not hold, when t-norm * and t-conorm ⊗ are
Conclusion
In this paper, we present the flaws in [14] with counterexamples. The calculations of T-extension operations of t-(co)norms and their residual operations are further investigated. The incorrect conclusions in [14] are corrected. Moreover, for the convenience of readers, we list a table summarizing the incorrect conclusions in [14] and the corresponding conclusions in our paper.
Acknowledgments
The author would like to thank the Editors and the anonymous reviewers for their valuable comments and suggestions in improving this paper.
References (25)
- et al.
Fuzzy decision making systems based on interval type-2 fuzzy sets
Inf. Sci.
(2013) - et al.
Interval type-2 fuzzy membership function generation methods for pattern recognition
Inf. Sci.
(2009) - et al.
Triangular norms on product lattices
Fuzzy Sets Syst.
(1999) - et al.
Type-2 fuzzy logic aggregation of multiple fuzzy controllers for airplane flight control
Inf. Sci.
(2015) - et al.
A type-2 fuzzy c-regression clustering algorithm for Takagi–Sugeno system identification and its application in the steel industry
Inf. Sci.
(2012) - et al.
Exact calculations of extended logical operations on fuzzy truth values
Fuzzy Sets Syst.
(2008) - et al.
On type-2 fuzzy sets and their t-norm operations
Inf. Sci.
(2014) - et al.
On type-2 fuzzy relations and interval-valued type-2 fuzzy sets
Fuzzy Sets Syst.
(2014) - et al.
Interval-valued possibilistic fuzzy C-means clustering algorithm
Fuzzy Sets Syst.
(2014) Notes on “exact calculations of extended logical operations on fuzzy truth values”
Fuzzy Sets Syst.
(2016)
Cited by (8)
An investigation of ambiguous sets and their application to decision-making from partial order to lattice ambiguous sets
2023, Decision Analytics JournalRevisiting type-2 triangular norms on normal convex fuzzy truth values
2023, Information SciencesDistributivity between extended t-norms and t-conorms on fuzzy truth values
2021, Fuzzy Sets and SystemsType-2 fuzzy implications and fuzzy-valued approximation reasoning
2018, International Journal of Approximate ReasoningCitation Excerpt :For example, Wang and Hu [38] proposed a fuzzy-valued fuzzy implication generated from a fuzzy-valued t-norm induced by a left-continuous t-norm. Li [25] discussed the residual operators of type-2 t-norms with respect to the partial order ⊑ induced by extended minimum, which were further investigated in [46]. Generalized extended fuzzy implications were investigated in accordance with the generalized extension principle in [39], where neither t-norms nor fuzzy implications are necessarily continuous.
On the Extensions of Overlap Functions and Grouping Functions to Fuzzy Truth Values
2021, IEEE Transactions on Fuzzy Systems