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This chapter describes the various aspects of triangular norm-based measures. Triangular norm-based measures (T-measures) are special real valuations defined on T-tribes, the latter being classes of fuzzy sets based on a triangular norm. The necessary preliminaries about triangular norms and fuzzy sets, including concepts of disjointness and emphasizing the special role of Frank t-norms are presented in the chapter. T-tribes are introduced and their most prominent properties are listed in the chapter. Three different representations for T-measures as integral with respect to a suitable Markov kernel, as well as a decomposition of monotone measures are presented in the chapter. The chapter discusses measures with respect to the Lukasiewicz t-norm and some of their most important properties, which include Jordan decomposition, absolute continuity, Darboux property, and nonatomicity. The Liapounoff type theorem concerning the compactness and convexity of the range of vector measures is presented in the chapter. The existence of an Aumann–Shapley value on the space of games with fuzzy coalitions spanned by positive integer powers of monotone TL-measures is also shown in the chapter.

Some properties of and relationships between important classes of copulas are discussed. In particular, the class of associative copulas is shown to be compact, and the classes of strict and non-strict copulas are shown to be dense subsets of this class.

This contribution presents a fuzzy method for a particular kind of pixel classification. It is one of the most important results of the development of an inspection system for a silk-screen printing process. The classification algorithm is applied to a reference image in the initial step of the printing process in order to obtain regions which are to be checked by applying different criteria. Tight limitations in terms of computation speed have necessitated very specific, efficient methods which operate locally. These methods are motivated and described in detail in the following. Furthermore, the optimization of the parameters of the classification system with genetic algorithms is discussed. Finally, the genetic approach is compared with other probabilistic optimization methods.

When we introduced several important families of t-norms in Chapter 4, we mentioned already (without proof) that all these families are continuous with respect to the parameter, i.e., that we have pointwise convergence of the t-norms if the corresponding parameters converge (some of these statements are trivial, some of them follow directly from [Dombi 1982, 1986]).

The theory of fuzzy sets was introduced in 1965 by Lotfi A. Zadeh as a generalization of the classical Cantorian set theory whose logical basis is the two-valued Boolean logic (compare also [Klaua 1965, 1967]). It was suggested in [Zadeh 1965] to use the minimum T M, the maximum S M, and the standard negation N S in order to model the intersection, the union, and the complement of fuzzy sets, respectively. However, also the product T p, the probabilistic sum S p and the Lukasiewicz t-conorm S L, (the latter in a restricted form) were already mentioned as possible candidates for the intersection and the union of fuzzy sets, respectively, in the very first paper on this subject.

The study of many-valued logic was initiated by Jan Lukasiewicz around 1920. He started with a three-valued logic, introducing in particular an implication for it (see [Lukasiewicz 1920, 1930] and [Lukasiewicz & Tarski 1930], a selection of Lukasiewicz’s papers can be found in [Borkowski 1970]). Another important step was the study of intuitionistic logic in [Heyting 1930, Godel 1932], significant contributions were also made in, e.g., [Post 1921], [Moisil 1940, 1941] and [McNaughton 1951, Chang 1958, Rose & Rosser 1958] (for surveys on many-valued logics we refer to the monographs [Rosser & Turquette 1952, Rescher 1969] and [Gottwald 1989, 200x]).

We start this overview with a discussion of the computational rule of inference [Zadeh 1973] which plays a crucial role in fuzzy control (see below) but also in approximate reasoning [Fuller & Zimmermann 1992]. It is a way to associate with a fuzzy input a fuzzy output by means of a fuzzy relation between the input and the output space. We also give a geometric and a logical interpretation of the computational rule of inference, the latter showing a close relationship to the R-implications studied in Section 11.1. If input and output are known, we present criteria for the solvability of the corresponding fuzzy relational equation (see [Sanchez 1974, 1976, 1977, 1984] and [Gottwald 1986, Di Nola et al. 1989]).

The aim of this chapter is to provide the reader with a collection of parameterized families of t-norms which we think are interesting from various points of view. We have chosen this compact form of presentation in order to simplify the search for specific examples.

Giving exactly this definition, Karl Menger in 1942 introduced triangular norms into the mathematical literature. Triangular norms (t-norms for short) were originally used when generalizing the triangle inequality from classical metric spaces to statistical metric spaces (today called probabilistic metric spaces, see Section 9.3) which were the key point of interest in [Menger 1942].

The full information about the values of a t-norm is given by the preimages of one-point sets {β} with β ∈ [0, 1] under the t-norm T or, equivalently, by the sets of solutions of the equations T(x, y) = β. For continuous t-norms these preimages are completely described in Section 7.1. For arbitrary t-norms, a complete characterization of the preimages of {0} is given. Also, for continuous t-norms T, the unique solvability of the equation T (a, x) = b with (a, b) ∈ [0,1[2 and b ≤ a is investigated.

Generalizing the inverse of a bijective function, we consider the pseudo-inverse of monotone functions from [0, 1] to [0, 1]. Introducing a construction similar to the one given in [Schweizer & Sklar 1983, Theorem 5.2.1] with the help of the so-called quasi-inverses, we state a rather general method to construct new t-norms from known t-norms using the pseudo-inverses of appropriate functions (compare also [Jenei 1999, Klement et al. 1999]). A number of examples (together with visualizations of prototypical cases) for this construction is provided.

From the many generalizations of the concept of a classical (σ-additive) measure which can be found in the literature we shall present two concepts where t-norms and t-conorms are used explicitly.

For the rather general class of all t-norms, which includes non-continuous t-norms and even t-norms which are not Borel measurable, no universal representation theorems exist so far. In fact, such a characterization of arbitrary t-norms would be closely related to the solution of the famous, still unsolved general associativity functional equation.

Triangular norms were introduced in the framework of probabilistic metric spaces [Schweizer and Sklar, 1961; Schweizer and Sklar, 1963; Schweizer and Sklar, 1983] , based on ideas first presented in [Menger, 1942] , and they are applied in several fields, e.g. in fuzzy sets [Zadeh, 1965] , fuzzy logics [Butnariu et al., 1995; Hajek et al., 1996; Pavelka, 1979] and their applications, but also in the theory of generalized measures [Butnariu and Klement, 1993; Klement and Weber, 1991] and nonlinear differential and difference equations [Pap, 1992].

The term ‘Fuzzy Logic’, although very well-known, is ambiguous as it refers to several only loosely related concerns ranging from rule-based system control to various multiple-valued logics. The most popular acception of the term ‘fuzzy logic’ is generally not related to logic proper, since it is primarily employed by control engineers that use fuzzy rule-based systems like a sort of neural network capable of approximating non-linear functions. However in the recent past, it has been stressed that fuzzy logic in the narrow sense can be envisaged from the point of view of logic provided that fuzzy sets are considered as stemming from the multiple-valued logic tradition. The relationship between multiple-valued logic and fuzzy sets had been noticed by Moisil [1972] in the late sixties. At the same period, in Eastern Germany, Klaua independently built up a multiple-valued set theory (see Gottwald [1984] for an extensive bibliography of Klaua’s papers). The current trend relating fuzzy sets and multiple-valued logic actually dates back to a seminal paper by Goguen [1969]. In this paper Goguen insists on an algebraic structure he calls a ‘closg’ (for commutative lattice ordered semi-group) and shows that the lattice-theoretical concept of residuation can be generalized to operations other than the minimum. Following Goguen’s program, Pavelka [1979] has definitely anchored fuzzy logic in the multiple-valued tradition, emphasizing a link with Lukasiewicz logic already pointed out by Giles [1976]. Since then, a large amount of work has been carried out whose aim is to equip fuzzy logic with a syntactic component, and several algebraic structures have been laid bare as potential candidates for supporting fuzzy logics.

Aufbauend auf den Vorschlagen von K. Menger [12], fuhrten B. Schweizer und A. Sklar [17] den Begriff der t-Normen (Dreiecksnormen) ein. Dieser diente dazu, in probabilistischen metrischen Raumen [17, 18] die von den klassischen Metriken wohlbekannte Dreiecksungleichung in geeigneter Weise zu verallgemeinern.

In this chapter, we try to present a coherent survey on some recent attempts in building a theory of generalized measures. Our main goal is to emphasize a minimal set of axioms both for the measures and their domains, and still to be able to prove significant results. Therefore we start with fairly general structures and enrich them with additional properties only if necessary.

In this paper a new approach for the coordination of the motion axes of a mobile manipulator based on fuzzy behavioral algorithms and its implementation on a physical demonstrator is presented. The kinematic redundancy of the overall system (consisting of a 7 DOF manipulator and a 3 DOF mobile robot) will be used for autonomous and reactive motion of the mobile manipulator within poorly structured and even dynamically changing surroundings. Sensors around the mobile and along the manipulator will provide the necessary information for navigation purposes and perception of the environment.

A fuzzy method for a particular kind of pixel classification is proposed. It is one of the most important results in the development of an inspection system for a silk-screen printing process. The classification algorithm is applied to a reference image in the initial step of the printing process in order to obtain the regions which are to be checked by applying different criteria. Tight limitations in terms of computation speed have necessitated very specific, efficient methods which operate locally. These methods are motivated and discussed in detail.

It is shown how fuzzy controllers, in particular the Mamdani and Sugeno controller, can be used to interpolate and approximate control functions, i.e., input-output functions which assign to each input value a real output value.

26.1 In this section we discuss some aspects of the problem of defining Aumann—Shapley values on spaces of games with crisp coalitions. By games with crisp coalitions we mean games defined on a σ-algebra A of crisp subsets of X (which is a particular T ∞ -tribe). As usual, we write BV and pNA for the spaces FBV A and pFNA A respectively. Also, putting ℐ = A^, we write FBV and pFNA instead of FBV ℐ and pFNA ℐ , respectively.

Theorem 18.4 shows that there always exists an Aumann-Shapley value on the space pFNA ℐ , no matter how the space X and the T∞-tribe. ℐ (generated or not) are chosen. Note that equation (18.6) in this theorem applies only to games of the form υ = f o m where m is a vector T∞ -measure and f : R(m) → ℝ is a continuously differentiable function such that f(0) = 0.

Fuzzy Mengen (im Deutschen etwa: unscharfe Mengen) wurden von ZADEH [1965] eingefuhrt. Ziel seiner bahnbrechenden Uberlegungen war es, eine Beschreibung realer Situationen zu ermoglichen, die der menschlichen Denkweise naher kommt als die klassische Cantor’sche Mengenlehre. Letztere geht bekanntlich davon aus, das von jedem Objekt x des vorgegebenen Universums U eindeutig gesagt werden kann, ob es in der betrachteten Menge liegt, also deren definierende Eigenschaft erfullt. Dies trifft sicherlich zu, solange man es mit so scharf begrenzten Mengen zu tun hat wie zum Beispiel bei der Menge aller Losungen der Gleichung x2 = 4.

The class of cooperative games, which is the main topic of this book and which we shall investigate in detail in chapters IV — VI, consists of games with fuzzy coalitions. In such coalitions players are allowed to belong in a gradual way (as opposed to the yes/no alternative in classical models of games in characteristic function form, as described by Neumann & Morgenstein [1944]). Fuzzy coalitions will be described by fuzzy sets [Zadeh 1965], a generalization of Cantorian sets. For a Cantorian set A the possible truth values of the statement “x is an element of A” are 0 and 1 (False and True), whereas for a fuzzy set A the truth values of the statement “x is an element of A” can be any number in the interval [0, 1]. We shall describe fuzzy sets in more detail in the next section; here we want to provide the tools for extending operations such as intersection and union of sets. This enables us to represent algebraically natural entropic rules for forming and dismantling fuzzy coalitions.

I: Topos-like and Model-Theoretic Approaches.- 1: Classification of Extremal Subobjects of Algebras over SM-SET.- 2: M-valued Sets and Sheaves over Integral Commutative CL-Monoids.- 3: The Logic of Unbalanced Subobjects in a Category with Two Closed Structures.- II: Categorical Methods in Topology.- 4: Fuzzy Filter Functors and Convergence.- 5: Convenient Topological Constructs.- 6: A Topological Universe Extension of FTS.- 7: Categorical Frameworks for Stone Representation Theories.- III: Applications and Related Topics in Logic and Topology.- 8: Pointless Metric Spaces and Fuzzy Spaces.- 9: Fuzzy Unit Interval and Fuzzy Paths.- 10: Lattice Morphisms, Sobriety, and Urysohn Lemmas.- 11: The Topological Modification of the L-Fuzzy Unit Interval.- 12: A Categorical Approach to Fuzzy Relational Database Theory.- 13: Fuzzy Points and Membership.- Appendices.- Index of Categories.- Addenda et Corrigenda.

We determine two constructions that, starting with two bivariate copulas, give rise to new bivariate and trivariate copulas, respectively. These constructions are used to determine pointwise upper and lower bounds for the class of all trivariate copulas with given bivariate marginals., The third author acknowledges the support by the Ministerio de Educación y Ciencia (Spain) and FEDER, under research project MTM2006-12218. This work has been partially supported by the bilateral cooperation Austria-Spain WTZ—“Acciones Integradas 2008/2009”, in the framework of the project Constructions of Multivariate Statistical Models with Copulas (Project ES04/2008).

Some experiences of the cooperation of the Fuzzy Logic Laboratorium Linz-Hagenberg (Austria) with industrial partners are described. Special emphasis is given to the intelligent combination of various mathematical methods.

Generalizing the definitions of L. A. Zadeh (J. Math. Anal. Appl. 23 (1968), 421–427) and E. P. Klement, R. Lowen, and W. Schwyhla (Fuzzy Sets and Systems, in press) a larger class of finite fuzzy measures is defined. It is shown that these fuzzy measures can be characterized in a unique way by a finite (classical) measure and a Markoff-kernel.

We show that slight technical changes in the definition transform the probability of fuzzy events introduced by R. R. Yager [16] into a new concept of such probabilities having nice properties, both from an intuitive and from a mathematical point of view: monotonicity, additivity, and continuity.

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra σ are discussed.

A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.