The structure of pan-addition operator with pre-determined pan-multiplication☆
Introduction
Due to the importance in fuzzy reasoning and decision-making, the study of T-operators has been one of the important topics in fuzzy mathematics. It can also be said that T-operators have made up of one of the main fundamental blocks in the research of fuzzy systems and fuzzy computations. In July 1993, during the 11th annual conference on artificial intelligence, held in Washington DC, there appeared a debate about fuzzy logic and relevant matters. This debate further clarified our understanding about fuzzy logic and fuzzy operators. Especially, Elkan [1] posted the well-known statement that it is universally true that there does not exist a function f such that t(A ∧ B) = f(t(A), t(B)). His example of watermelon “evidence intensity” has well explained why min operator is not appropriate in this special situation. It provides a need for us to further study other operators. The concept of T-operators has been generalized to pan-arithmetic, pan-operators, semi-rings, etc. In the earlier stage of research along this line, scholars had considered a series of T-operators and their properties [3], [4], [5], [6], [7]. For example, Sugeno and Murofushi [9], [10] obtained results regarding the structure of quasi-operators under certain conditions. Wang and Klir [13] generalized the concept of triangular modules to that of pan-operators. The structure of pan-operators under “⊕ ≠ ∨” has been considered by Mesiar and Rybarik [5]. Tong et al. [11] listed some counterexamples to show that these results are incorrect in terms of fuzzy sets and systems and new and modified results were given. As for the structure of quasi-operators under “⊕ = ∨”, [8] did not provide anything. As for the structure of pan-operators under the same condition, in 1995 Mesiar and Rybarik [5] offered a simple conjecture for the case of fuzzy sets and systems as an open problem. Tong et al. [12], [14] had settled this open problem by providing a structure for the general pan-multiplication operators. All the current results are obtained by studying the structures of pan-multiplication and pan-addition operators with pre-determined pan-addition operators. Now, if one considers the opposite, if a pan-multiplication operator is given, he would be able to ask: Is the pan-addition operator, which forms a commutative isotonic semi-ring with the given pan-multiplication operator, unique? If the existence of the pan-addition operator is not unique, how many are there? What is the structure? In this paper, we will provide answers to address all these problems.
Section snippets
Basic definitions
The following is the axiomatic definition of pan-operators given by [13]. Definition 2.1 Wang and Klir A binary operation ⊕, defined on the non-negative half of the real number line , is termed to as a pan-addition, iff it satisfies the following axioms: x ⊕ y = y ⊕ x (commutative law); (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z) (associative law); for any , x ⩽ y ⇒ x ⊕ z ⩽ y ⊕ z (law of monotonity); for any , x ⊕ 0 = x (existence of zero element); and limn→∞xn = x, limn→∞yn = y ⇒ limn→∞(xn ⊕ yn) = x ⊕ y (law of continuity).
Definition 2.2 Wang and Klir
Assume that e and ⊕ are a binary operation and a
The structure of pan-addition with pre-determined pan-multiplication
Theorem 3.1 If the triple is a commutative isotonic semi-ring with the unit element e not being exponentially equal, then there is a maximum exponentially idempotent element a and an idempotent element b with respect to the binary operation ⊕ on [0, e) and a strictly monotonic increasing continuous function g such thatand that for any x, y ∈ [0, +∞], for the following holds true:and[11]
Summary
Applying results from literature [11], we have considered the structure of any pan-addition with pre-fixed pan-multiplication. For the special case, i.e. when the pan-multiplication is the ordinary multiplication, by employing the distributive law of the ordinary multiplication over commutative isotonic semi-ring, we obtained a differential equation, which the function derived through the pan-addition satisfies. Through solving this differential equation, we have obtained the structure of the
References (14)
A note on the idempotent functions with respect to pseudo-convolution
Fuzzy Sets and Systems
(1999)- et al.
PAN-operations structure
Fuzzy Sets and Systems
(1995) - et al.
Shape preserving additions of fuzzy intervals
Fuzzy Sets and Systems
(1997) Connective generators for Archimedian triangular operators
Fuzzy Sets and Systems
(1998)- et al.
Pseudo-additive measures and integrals
Journal of Mathematical Analysis and Applications
(1987) - et al.
Pan-operations structure with non-idempotent pan-addition
Fuzzy Sets and Systems
(2004) The paradoxical success of fuzzy logic
IEEE Expert
(1994)
Cited by (3)
SF-FCM based on fast close function and subtractive clustering
2011, Kongzhi yu Juece/Control and DecisionTwo-stage fuzzy c-mean cluster and its applications
2008, Huazhong Keji Daxue Xuebao (Ziran Kexue Ban)/Journal of Huazhong University of Science and Technology (Natural Science Edition)The structure of pan-operator with idempotent pan-addition
2007, Proceedings of the Sixth International Conference on Machine Learning and Cybernetics, ICMLC 2007
- ☆
This research is partially supported by National Natural Science Foundation (79970025, 69874018) and the National Defense Science and Technology (OOJ15.3.JWO528).
- 1
Tel.: +86 2762097718/2783937411.