Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms (eBook)

Erich Peter Klement, Radko Mesiar (Herausgeber)

eBook Download: PDF

2005 | 1. Auflage
492 Seiten
Elsevier Science (Verlag)
978-0-08-045953-0 (ISBN)

Infos

This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16 chapters written by leading experts provide a state of the art overview of theory and applications of triangular norms and related operators in fuzzy logic, measure theory, probability theory, and probabilistic metric spaces.

Key Features:

- Complete state of the art of the importance of triangular norms in various mathematical fields
- 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications
- Chapter authors are leading authorities in their fields
- Triangular norms on different domains (including discrete, partially ordered) are described
- Not only triangular norms but also related operators (aggregation operators, copulas) are covered
- Book contains many enlightening illustrations

? Complete state of the art of the importance of triangular norms in various mathematical fields
? 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications
? Chapter authors are leading authorities in their fields
? Triangular norms on different domains (including discrete, partially ordered) are described
? Not only triangular norms but also related operators (aggregation operators, copulas) are covered
? Book contains many enlightening illustrations

Inhalt

front cover 1
copyright 5
table of contents 6
front matter 8
Preface 8
body 10
I: INTRODUCTION 10
1 Triangular norms, looking back-triangle functions, looking ahead 12
2 Triangular norms: Basic notions and properties 26
II: THEORETICAL ASPECTS OF TRIANGULAR NORMS 70
3 Semigroups and triangular norms 72
4 Generators of triangular norms 104
5 A survey on left-continuous t- norms and pseudo t-norms 122
6 Some aspects of functional equations 152
7 Triangular norms on discrete settings 198
8 Triangular norms and related operators in L*- fuzzy set theory 240
9 Fitting triangular norms to empirical data 270
III: APPLICATIONS OF TRIANGULAR NORMS AND RELATED OPERATIONS 282
10 Triangular norm- based mathematical fuzzy logics 284
11 Many-valued equalities and their representations 310
12 Varieties of algebras in fuzzy set theory 330
13 Triangular norms and measures of fuzzy sets 354
14 Copulas and quasi- copulas: An introduction to their properties and applications 400
15 Transitive comparison of random variables 424
16 Triangular norms in probabilistic metric spaces and fixed point theory 452
Index 482

Textauszug

1

Triangular norms, looking back—triangle functions, looking ahead

Berthold Schweizer [e-mail]    Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA (USA)

1.1

In [1942], in a short note entitled “Statistical Metrics”, Karl Menger introduced the notion of a space in which distances are determined by probability distribution functions rather than by real numbers. To extend the triangle inequality to this setting, he employed a function T from I2, the closed unit square, to I the closed unit interval, which he called a triangular norm. Specifically, he assumed that for all a, b, c, d in I,

Tab=Tba,Tab≤Tcdwhenevera≤candb≤d,T11=1andTa1>0whenevera>0.

Soon thereafter, Abraham Wald [1943] suggested a different triangle inequality, which Menger adopted. Subsequent developments have shown that Wald’s inequality is too strong; and because of this Menger was not able to make significant headway in the subject.

In the mid-1950’s, Abe Sklar and I began our collaboration on the theory of probabilistic (née statistical) metric spaces. Our “breakthrough” came in [1958] when we returned to Menger’s triangle inequality. Elementary considerations showed that the third of Menger’s conditions should be replaced by the stronger condition: T(a, 1) = a. But more important, in order to extend his triangle inequality to a polygonal inequality, we stipulated that T be associative. Thus our requirements on T became: For all a, b, c, d in I,

(a) T(a, 1) = a,

(b) T(a, b) = T(b, a),

(c) T(a, b) ≤ T(c, d) whenever a ≤ c and b ≤ d,

(d) T(T(a, b),c) = T(a, T(b, c)).

We called a function T satisfying the conditions (a)–(d) a t-norm.

At about this same time, Sklar, in answer to a question of M. Fréchet, introduced the notion of a copula and proved the important theorem that now bears his name. (See R. B. Nelsen’s contribution to this volume for definitions, details and references.)

Menger’s triangle inequality states that

prx+y≥TFpqx,Fqry,

  (1.1)

for all p, q, r and all x, y. Here p.q.r are points in the underlying space, x, y are real numbers, Fpr, Fpq, Fqr are the probability distribution functions associated with the pairs {p, r}. {p, q}, {q, r}, respectively, and T is a t-norm. Since the structural properties of the probabilistic metric space depend on T, it is desirable to have a repertoire of t-norms at one’s disposal. In our paper [1961] Sklar and I addressed this problem:

In the first section we pointed out that the functional equation of associativity (d) was first solved, under commutativity and differentiability assumptions, by Abel in 1826, in the very first paper that he published in Crelle’s Journal (see [Lawson, 1996] for a detailed discussion) and we presented J. Aczél’s fundamental representation theorem for such functions [Aczél, 1949, 1966]. At the time we were unaware of the fact that Aczél’s result is a partial solution of the second part of Hilbert’s Fifth Problem [Aczél, 1989].

In the second section we adapted Aczél’s theorem to our purpose and showed that every strict (i.e., strictly increasing on (0,1] × (0,1] and continuous) t-norm T admits the representation

ab=f−1fa+fb,

  (1.2)

for all a, b in I, where the function f : I → [0, ∞] is continuous and strictly decreasing with f(0) = +∞ and f(1) = 0, and f–1 is the inverse of f. We called f an additive generator of T and showed that any two such generators differ by a multiplicative constant. We also introduced multiplicative generators and showed that any strict t-norm T is multiplicatively generated from the t-norm P given by P(x, y) = xy via

xy=h−1Phx,hy,

  (1.3)

where the function h: I → I is continuous and strictly increasing with h(0) = 0 and h(1) = 1, and h–1 is the inverse of h.

The representation (1.2) shows that strict t-norms—indeed entire families of such t-norms—are as easy to generate as continuous strictly decreasing functions from I to [0,∞]. In the third section we exploited this fact to construct several one-parameter families of strict t-norms. In recent years, this procedure has become something of a cottage industry and there are now probably more than enough such families to satisfy anyone’s wants (see [Klement, Mesiar and Pap, 2000] and [Alsina, Frank and Schweizer, 200x]).

In the fourth section we looked at the surface z = T(x, y) and gave the (obvious) interpretations of the conditions (a), (b) and (c). This is where the standard picture of the “frame” determined by the boundary conditions T(a, 1) = T( 1, a) = a and T(a, 0) = T(0, a) = 0 for all a in I made its first appearance. We also asked for geometric interpretations of the associativity condition (d). Our conjecture that every sufficiently smooth “associative surface” is isothermal is still not settled.

In Section 5 we applied the representation (1.2) to Menger’s triangle inequality (1.1). We also showed that if T1 and T2 are strict t-norms with additive generators f1 and f2, respectively, then T1 is weaker than T2 (i.e., T1 (a, b) ≤ T2(a, b) for all a, b in I and T1 ≠ Τ2) if and ony if the composite function f1of− 12 is non-linear and subadditive.

In Section 6, we introduced t-conorms and worked out the simple arithmetic and geometric relationships between them and t-norms. (Since these functions often arise in their own right and not as the conorm of some t-norm, i.e., not as 1 – T(1 – x, 1 – y), I now prefer the term s-norm for any function S satisfying (b), (c), (d) and the boundary condition S(a, 0) = a; but for the sake of consistency with the other contributions to this volume, I will refrain from making this change here.)

In the final section we proved that a strict t-norm satisfies the inequality

ad+Tcb≤Tab+Tcd

for all a, b, c, d in I such that a ≤ c and b ≤ d if and only if any additive generator of T is convex. From this we immediately deduced the important fact that a strict t-norm T is a (two-dimensional) copula if and only if any additive generator of T is convex.

The paper closed with a brief discussion of the relationship between copulas and the joint distributions of pairs of distances in a probabilistic metric space.

In retrospect, 40-plus years later, it is remarkable how many of the basic elements of the theory of t-norms, t-conorms and copulas were introduced in this paper. Virtually all of the results later found their way into our book [Schweizer and Sklar, 1983]; and most of them are reproduced in every introduction to t-norms and their properties.

In a subsequent paper Sklar and I [1963] continued our study of t-norms. We showed how (1.3) could be used to generate not necessarily strict t-norms from t-norms other than...

Produkt-Details

• Erscheint lt. Verlag: 25.3.2005
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• ISBN-10: 0-08-045953-6 / 0080459536
• ISBN-13: 978-0-08-045953-0 / 9780080459530