Problems on associative functions. (English) Zbl 1077.39021

Alsina, C.; Frank, M. J.; Schweizer, B.

Aequationes Math. 66, No. 1-2, 128-140 (2003).

A function \(F:I \times I \to I\), where \(I\) is a real interval, is said to be associative if the following relation holds: \[ F(F(x,y),z)=F(x,F(y,z)), \quad x,y,z \in I. \] A special class of associative functions, the so-called \(t\)-norms, appears when studying probabilistic metric spaces. A \(t\)-norm is a function \(T\) from the unit square to the unit interval, such that for all \(x,y,z,w \in [0,1]\), \[ \begin{aligned} &T(x,1)=x, \tag{a}\\ &T(x,y)=T(y,x), \tag{b}\\ &T(x,y) \leq T(z,w) \;\text{ whenever }\;x\leq z, \;y\leq w, \tag{c}\\ &T(T(x,y),z)=T(x,T(y,z)).\tag{d} \end{aligned} \] The authors studied extensively probabilistic metric spaces [see the book of B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics. New York-Amsterdam-Oxford: North-Holland. (1983; Zbl 0546.60010)] and consequently \(t\)-norms and related notions. In the present survey paper, after the presentation of the basic definitions, they propose 21 problems concerning associative functions. The paper contains a rich bibliography on the subject.

Reviewer: Gian Luigi Forti (Milano)

MSC:

39B22	Functional equations for real functions
39B52	Functional equations for functions with more general domains and/or ranges
39-02	Research exposition (monographs, survey articles) pertaining to difference and functional equations
39B62	Functional inequalities, including subadditivity, convexity, etc.
62H05	Characterization and structure theory for multivariate probability distributions; copulas
60B05	Probability measures on topological spaces

Keywords:

Associative functions; \(t\)-norms; \(s\)-norms; Copulas; Strong negations; Open problems; Probabilistic metric spaces; Survey paper; Bibliography