Endre Pap

Triangular norms. Position paper I: basic analytical and algebraic properties

We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan triples are also mentioned. Finally, a brief historical survey on triangular norms is given.

A Universal Integral as Common Frame for Choquet and Sugeno Integral

The Choquet and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. However, their applicability is restricted because of the special operations used in the construction of these integrals. Therefore, we provide a concept of integrals generalizing both the Choquet and the Sugeno case. For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated, which can be defined on arbitrary measurable spaces and for arbitrary monotone measures. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced, and the restriction to the unit interval, i.e., to fuzzy integrals, is considered.

On the relationship of associative compensatory operators to triangular norms and conorms

When using a t-norm for combining fuzzy sets, no compensation between small and large degrees of membership takes place. On the other hand, a t-conorm provides full compensation. Since many real situations do not fall into either one category, so-called compensatory operators have been proposed in the literature [H.-J. Zimmermann and P. Zysno, Fuzzy Sets and Systems4 (1980) 37–51] which are non-associative in nature. In this paper, associative compensatory operators (whose domain is the unit square with the exception of the two points (0, 1) and (1, 0) and whose only associative extensions to the whole unit square are the aggregative operators suggested in [J. Dombi, Europ. J. Oper. Res.10 (1982) 282–293]) are studied and their representation in terms of multiplicative generators is given. It is shown that these operators are constructed with the help of strict t-norms and t-conorms, in a way which is similar to ordinal sums. Finally, the duals of such operators are shown to be again associative compensatory operators, and a characterization of self-dual operators is given.

Aggregation functions: Means

This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means).

CHAPTER 35 – Pseudo-Additive Measures and Their Applications

This chapter discusses the pseudoadditive measures and and the corresponding integrals, which give a base for pseudoanalysis. The pseudoadditive measures are applied in optimization problems, nonlinear partial differential equations, nonlinear difference equations, optimal control, and fuzzy systems.Pseudoanalysis uses many mathematical tools from different fields, such as functional equations, variational calculus, measure theory, functional analysis, optimization theory, and semiring theory. The advantage of the pseudoanalysis is that the problems from many different fields are covered with one theory and unified methods. This approach gives solutions in such a form that are not achieved by other theories. In some cases, it enables nonlinear equations to obtain exact solutions in the similar form as for linear equations. Some obtained principles such as the pseudolinear superposition principle allows transferring methods of linear equations to nonlinear equations. Pseudointegral that is defined as the limits of the corresponding idempotent Riemannian sums is also elaborated in the chapter.

Triangular norms. Position paper II: General constructions and parameterized families

This second part (out of three) of a series of position papers on triangular norms (for Part I see Triangular norms. Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems, in press) deals with general construction methods based on additive and multiplicative generators, and on ordinal sums. Also included are some constructions leading to non-continuous t-norms, and a presentation of some distinguished families of t-norms.

Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms

Abstract Several properties of quasi- and pseudo-inverses of a non-decreasing real function are discussed. Based on a result of Schweizer and Sklar, for a given triangular norm T and non-decreasing function f a construction method leading to a commutative, fully ordered semigroup on the unit interval is given. A similar construction based on the pseudo-inverse implies that the resulting operation will be bounded from above by the minimum, but then the associativity may be violated. Several sufficient conditions for constructing new t-norms from a given one and a non-decreasing function f , based on its quasi-inverses and on its pseudo-inverse, respectively, are discussed, together with illustrative examples.

Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes

In this second part of our state-of-the-art overview on aggregation theory, based again on our recent monograph on aggregation functions, we focus on several construction methods for aggregation functions and on special classes of aggregation functions, covering the well-known conjunctive, disjunctive, and mixed aggregation functions. Some fields of applications are included.

Idempotent integral as limit of g -integrals

We show that sup- and inf-decomposable measures can be obtained as limits of families of pseudo-additive measures with respect to generated pseudo-additions. The corresponding integrals with respect to sup- or inf-decomposable measures can be obtained as limits of families of g-integrals.

Triangular norms. Position paper III: continuous t-norms

This third and last part of a series of position papers on triangular norms (for Parts I and II see (E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems, in press; E.P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper II: general constructions and parameterized families, submitted for publication) presents the representation of continuous Archimedean t-norms by means of additive generators, and the representation of continuous t-norms by means of ordinal sums with Archimedean summands, both with full proofs. Finally some consequences of these representation theorems in the context of comparison and convergence of continuous t-norms, and of the determination of continuous t-norms by their diagonal sections are mentioned.