On type-2 fuzzy relations and interval-valued type-2 fuzzy sets
Introduction
Type-2 fuzzy sets (T2 FSs)—that is, fuzzy sets with special fuzzy sets as truth values—were introduced by Zadeh [50] in 1975 as an extension of the concept of type-1 fuzzy sets (T1 FSs), i.e., ordinary fuzzy sets defined in [49]. T2 FSs were explored and equivalently expressed in [2], [16], [18], [19], [24], [26], [27], [29], [37], [40]. T2 FSs had already been used in many areas [6], [17], [24], [26], [27], [28], [29], [36], [41], [42], [43].
The operations on T2 FSs were discussed in [3], [4], [7], [8], [10], [11], [13], [14], [15], [16], [17], [18], [19], [20], [21], [23], [37], [39], [46], [47], especially in [16], which presented a systematic investigation into t-norm extended operations between fuzzy truth values for T2 FSs. In [18], efficient algorithms for operations on T2 FSs had been presented, which were recursively generalized to type-n fuzzy sets in [1]. But the requirements of algorithms in [18] are too restrictive, for instance, fuzzy truth values must attain 1 at some point of the unit interval. In this paper, we discuss a more generalized situation where the least upper bounds of fuzzy truth values attain 1. Moreover, the previously introduced t-norm extended operations only deal with operations combining a finite number of fuzzy truth values. A new operation combining an infinite number of fuzzy truth values is needed to investigate the properties of T2 FSs. This is the first motivation of this research, that is, to present the definitions and properties of extended supremum and extended infimum.
As is well known, fuzzy relations play an important role in the theory of T1 FSs and their different kinds of applications. Type-2 fuzzy relations (T2 FRs) must be studied, which were introduced in [18], because a new type of fuzzy sets was introduced. The other motivation of this research is to investigate T2 FRs and their compositions.
In [16], Hu and Kwong had pointed out that the interval type-2 fuzzy sets, which were used in [23], [25], [26], [30], [31], [32], [33], [34], [35], [48], can be viewed as interval-valued fuzzy sets. Inspired by it, the interval-valued type-2 fuzzy sets (IVT2 FSs) and interval-valued type-2 fuzzy relations (IVT2 FRs) should be properly proposed, which are special classes of type-3 fuzzy sets. Instead of discussing type-3 fuzzy sets, the operations on IVT2 FSs and compositions of IVT2 FRs are investigated in this paper, which are more intuitive and simpler than those in [1]. It is another motivation of this research.
In Section 2, we recall some fundamental concepts and related properties of t-norm extended operations between fuzzy truth values. In Section 3, we introduce new operations, extended supremum and extended infimum, and discuss their properties. In Section 4, the main part of this paper, we systematically investigate the properties of T2 FSs and their compositions. In Section 5, we propose and study IVT2 FSs and IVT2 FRs. Moreover, some examples are presented. In the final section, our researches are concluded.
Section snippets
Preliminaries
Let X and Y be nonempty sets, which are said to be universes, and be the set of all mappings from X to Y. Henceforth, or , where for all . J denotes either a linearly ordered set with an involution negator N or the associated algebra , where N is decreasing and satisfies for all . A closed interval on J is a set of the form , and denoted as with . If J is bounded, then the smallest and greatest
Extended supremum and extended infimum
Only operations combining a finite number of fuzzy truth values are considered in Definition 2.3. In this section, we propose two operations combining an infinite number of fuzzy truth values. Definition 3.1 Let for all . Then we define
It is clear that for crisp sets for all ,
Let for all and Λ be
On type-2 fuzzy sets and their operations
Type-2 fuzzy sets were investigated in [2], [16], [18], [19], [24], [26], [27], [29], [37]. For convenience of readers, we give a brief introduction to type-2 fuzzy sets in the following.
Let X be a nonempty universe. Then a mapping is called a type-2 fuzzy set (T2 FS, for short) on X, where , called a fuzzy grade [38], which is a fuzzy truth value on J for all with membership function for all . The set of all type-2 fuzzy sets on X is denoted by
Definition of interval-valued type-2 fuzzy sets
Definition 5.1 A mapping is called an interval-valued type-2 fuzzy set (IVT2 FS, for short) on X. We denote with and , where and are said to be a lower T2 FS and an upper T2 FS about , respectively. The set of all IVT2 FSs on X is denoted by .
The lower and upper footprints of uncertainty (LFOU and UFOU, for short) are the footprints of uncertainty of and , respectively, i.e.,
Conclusion
In this paper, extended supremum and extended infimum are proposed and investigated. On this basis, this paper presents the properties of type-2 fuzzy relations and their compositions. It is easy to show that the properties of type-2 convex normal fuzzy relations are similar to the properties of fuzzy relations, when the universes are finite. However, they are completely different from each other when the universes turn to be infinite. It is a specific topic to study the properties of type-2
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and suggestions. This research was supported by the National Nature Science Foundation of China (Grant No. 61179038).
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