Elsevier

Information Sciences

Volume 176, Issue 3, 6 February 2006, Pages 321-331
Information Sciences

The structure of pan-addition operator with pre-determined pan-multiplication

https://doi.org/10.1016/j.ins.2004.10.007Get rights and content

Abstract

Based on the structure of general pan-multiplication operators and the theory of first order linear partial differential equations, we study the structure of pan-addition operators with pre-determined pan-multiplication operators. At first, we describe the structure of pan-addition operators under the ordinary multiplication. Then, with this structure in place, we derive the general representation for the structure of pan-addition operators.

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 R¯+, is termed to as a pan-addition, iff it satisfies the following axioms:

  • (PA1)

    x  y = y  x (commutative law);

  • (PA2)

    (x  y)  z = x  (y  z) (associative law);

  • (PA3)

    for any zR¯+, x  y  x  z  y  z (law of monotonity);

  • (PA4)

    for any xR¯+, x  0 = x (existence of zero element); and

  • (PA5)

    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

[11]

If the triple (R¯+,,) 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 operationon [0, e) and a strictly monotonic increasing continuous function g such thatg:[a,+)[0,+),g(a)=0,g(+)=+and that for any x, y  [0, +], for the following holds true:xy=g-1(g(x)+g(y))ifx,yaxyotherwiseandxy=g-1(g(x)g(y))ifx,ya{(ak,bk),gk,:kK}ifx,yax

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

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Cited by (3)

This research is partially supported by National Natural Science Foundation (79970025, 69874018) and the National Defense Science and Technology (OOJ15.3.JWO528).

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