Generalized extended fuzzy implications
Introduction
Fuzzy implications are the main operations in fuzzy logic [17], generalizing the classical implications to fuzzy logic. Different fuzzy implications have constructed various structures of fuzzy logic systems, such as the basic logic [17] which was shown to be the logic of continuous t-norms and their R-implications [7], the monoidal t-norms based logic [11] which was shown to be the logic of lower semicontinuous t-norms and their R-implications and so on. Moreover, they had been extended to type-2 fuzzy sets [49] in [15] and type-2 fuzzy logic systems were proposed in [25] from the view of type reductions and centroids [27]. However, it had been pointed out in [15] that there do not exist adjoint pairs between extended t-norms and extended fuzzy implications.
Type-2 fuzzy sets were introduced by Zadeh in [49]. Type-2 fuzzy sets are special fuzzy sets equipped with fuzzy truth values mapping from the unit interval to itself. As a consequence, fuzzy sets and interval-valued fuzzy sets [43], [49] are special cases of type-2 fuzzy sets, which were equivalently expressed [1], [26], [27], [33], [37], [39] and had already been used in many aspects [8], [24], [33], [34], [35], [36], [37], [38], [42], [44], [45].
Operations on type-2 fuzzy sets were investigated in many literatures [3], [4], [9], [10], [12], [13], [14], [15], [18], [19], [20], [21], [22], [23], [24], [26], [28], [29], [32], [39], [41], [46], [47], [48], such as extended t-(co)norms and extended fuzzy implications in accordance with Zadeh's extension principle. In [15], Gera and Dombi proposed type-2 fuzzy implications and investigated extended operations, where fuzzy implication was limited to be S-implication (resp. R-implication) based on maximum (resp. minimum). The generalized extended operations were studied in [22], [23], [29], [41] in accordance with generalized extension principle [49], where t-norms were supposed to be continuous. However, continuous t-norms are too special. Considering fuzzy implications do not satisfy associativity, we shall discuss here lower and upper semicontinuous t-norms. In this paper, we investigate generalized extended fuzzy implications on different algebras of fuzzy truth values, where t-norms are not necessarily continuous. However, when generalized extended fuzzy implications are closed on the algebra of convex fuzzy truth values, fuzzy implications need to be continuous [15]. Thus continuity of fuzzy implications is required, while generalized extended fuzzy implications are type-2 fuzzy implications on the algebra of convex normal fuzzy truth values. Since a fuzzy value can be equivalently represented by a nest of closed intervals, we shall express generalized extended fuzzy implications in the form of nests of closed intervals on the algebra of fuzzy values. On this basis, fuzzy-valued fuzzy implications induced by arbitrary fuzzy implications and t-norms are proposed on the algebra of fuzzy values, regardless of the continuity of fuzzy implications, and the lower (resp. upper) semicontinuity of t-norms. Furthermore, a fuzzy-valued fuzzy implication is identical with a generalized extended fuzzy implication on the algebra of fuzzy values, when they are induced by exactly the same continuous fuzzy implication and same upper semicontinuous t-norm.
The contents of the paper are organized as follows. In Section 2, we recall some fundamental concepts and related properties. In Section 3, we investigate generalized extended fuzzy implications on different algebras of fuzzy truth values. Especially, we study the sufficient and necessary conditions when a generalized extended fuzzy implication is a type-2 fuzzy implication. Section 4 discusses generalized extended fuzzy implications on the algebra of fuzzy values. Moreover, we construct fuzzy-valued fuzzy implications from arbitrary fuzzy implications and t-norms, regardless of the continuity of fuzzy implications, and the lower (resp. upper) semicontinuity of t-norms. In the final section, our research is concluded.
Section snippets
Preliminaries
Let X and Y be nonempty universes, be the set of all mappings from X to Y and I denote the unit interval . is denoted as , if . A binary operation (resp. ) is called a t-norm (resp. t-conorm) on I if it is commutative, associative, increasing in each argument and has a unit element 1 (resp. 0).
An involutive negator is a decreasing function and satisfies for all . Assume that ⁎ and ⋆ are two binary operations on I, then they are said to
Generalized extended fuzzy implications
Some basic properties of generalized extended operations are presented in the following proposition.
Proposition 3.1 Let and ⁎ be a binary operation on I. Then , if operations ⁎ and ⋆ are dual w.r.t. N on I. . and for all . The following hold for all , , if for all ; , if for all ; , if for all ; , if for all .
Proof (1) For all ,
Fuzzy-valued fuzzy implications
In this section, generalized extended fuzzy implications are studied on the algebra of fuzzy values and fuzzy-valued fuzzy implications are constructed from arbitrary fuzzy implications and t-norms.
A nest of closed intervals H can represent a fuzzy value, where H satisfies is a closed interval on I for all . The family of all nests of closed intervals on I is denoted as .
The following properties are very useful in studying generalized extended fuzzy implications.
Lemma 4.1 Let and
Conclusion
On the basis of the properties of fuzzy implications and t-norms, generalized extended fuzzy implications are investigated on different algebras of fuzzy truth values. The sufficient or necessary conditions are studied, when a generalized extended fuzzy implication is a type-2 fuzzy implication on the algebra of convex normal fuzzy truth values (resp. fuzzy values). On this basis, fuzzy-valued fuzzy implications induced by arbitrary fuzzy implications and t-norms are constructed by nests of
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant No. 61179038).
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