On type-2 fuzzy sets and their t-norm operations
Introduction
The concept of a type-2 fuzzy set (T2 FS)—that is, fuzzy set with fuzzy sets as truth values—was introduced by Zadeh in 1975 [56] as an extension of the concept of an ordinary fuzzy set [henceforth called a type-1 fuzzy set (T1 FS)] and equivalently expressed by Mendel [29], Karnik and Mendel [23], [24], and Mendel and John [34]. Overviews on T2 FSs were presented by Mendel in 2007 [31], [32]. T2 FSs have already been used in many aspects [4], [5], [22], [31], [32], [33], [34], but in most theories and applications, interval type-2 fuzzy sets, a special case of T2 FSs, were used [4], [5], [26], [30], [31], [35], [36], [37], [38], [39], [40], [41], [52], [58]. In fact, an interval type-2 fuzzy set is an interval-valued fuzzy set (IVFS). Since the term “interval type-2 fuzzy set” easily leads to misunderstanding, the interval-valued type-2 fuzzy set (IVT2 FS), a fuzzy set with interval-valued fuzzy sets as truth values, should be introduced properly. With regards to this, further studies have to be done on type-2 fuzzy sets, and their operations based on the IVT2 FS.
The operations of T2 FSs, which are defined by t-norm, are main research topics [1], [2], [14], [15], [16], [18], [26], [42], [43], [44], [46], [47], [48], [49], [50], [51] especially in early literature [42], [43]. The truth value algebra of T2 FS is its operation base. This is another motivation of this research that is to extend operations of fuzzy true values to general binary operation on a linearly ordered set and triangle norms.
In Section 2, we discuss T-extension operations of fuzzy sets on a linearly ordered set, with the unit interval and real number set as special cases. Extended minimum and maximum, extended t-norm and t-conorm are studied on a linearly ordered set. In Section 3, studies of normal and T-convex fuzzy sets, and fuzzy numbers are described. In Section 4, T2 FSs are modified in their expressions and extended on their operations. In this section, two fuzzy sets (lower embedded FS and upper embedded FS) and maximal embedded IVFS on T2 FS are introduced. At the end of this section, we introduce type-2 fuzzy numbers (T2 FNs) and discuss their properties. In the final section, Section 5, conclusions on our research are given.
Section snippets
T-extension operations of fuzzy sets
Let X and Y be nonempty sets, which are referred to as universes, and Map(X, Y) be a set of all functions from X into Y. Henceforth, I = [0, 1] or I = ([0, 1], ∨ , ∧ , c, 0, 1), where ∧, ∨, and c are defined byJ denotes either a linearly ordered set with an involution N (i.e., N(N(x)) = x, ∀x ∈ J), or the associated algebra (J, ∨, ∧, N). We show N(K) = {N(r):r ∈ K} for K ⊆ J. If for any A ⊆ J, inf A ∈ J and sup A ∈ J as the linear order, then J is called complete. If J is bounded, then the
T-convex fuzzy sets and fuzzy numbers on fuzzy true values
To study convexity and fuzzy numbers for type-2 fuzzy sets, we first must study convex fuzzy sets and fuzzy numbers on fuzzy true values.
Definitions of type-2 fuzzy sets
The following definition of type-2 fuzzy sets is introduced before discussing their operations. Definition 4.1 A type-2 fuzzy set (T2 FS) of X, denoted as , is a mappingi.e., , where , called a fuzzy grade [43], is an ordinary fuzzy set on J for all x ∈ X with membership function . Ax is called the secondary membership function (MFnd); Ax (r) is called the secondary membership grade (MGnd); the support set of the MFnd, i.e.,
Conclusions
Type-2 fuzzy set is a “fuzzy-fuzzy set”, in which it is very important to fuzzy true values. This paper has studied the operations of fuzzy true values on a linearly ordered set J, with a unit interval and a real number set as special cases. The following are main conclusions.
- (1)
T-extension operations of general operator * on J are defined, and their properties are discussed, especially for some special cases, such as and .
- (2)
Normal fuzzy sets, T-convex fuzzy sets and fuzzy numbers on J
Acknowledgments
The authors thank the anonymous reviewers and the Editor-in-Chief, Professor Witold Pedrycz for their valuable suggestions in improving this paper. The work described in this paper was jointly supported by grants from The Hong Kong Polytechnic University (Project No. G-YK06) and The National Natural Science Foundation of China (Grant No. 61179038).
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