##
**Associative functions. Triangular norms and copulas.**
*(English)*
Zbl 1100.39023

Hackensack, NJ: World Scientific (ISBN 981-256-671-6/hbk). xiv, 237 p. (2006).

This well written and very readable book is probably quite unique: its main subject is a single functional equation, that of associativity (joined by some more or less related equations) and several of its applications. While coming from algebra, here it is applied mainly to probabilistic metric spaces (those used to be called statistical metric spaces) in particular to t-norms (those used to be called triangular norms; in this book that name shows up only in its subtitle and on p. 6, where the name-change is explained) and copulas. Therefore it is used in this book, as customary, in the analytic form \( T[T(x,y),z]=T[x,T(y,z)]\) rather than the algebraic one \( (x*y)*z=x*(y*z).\)

We will, however, define here t-norms and copulas using the latter notation: A t-norm \(* : [0,1]^2\to[0,1]\) is an associative (\((x\ast y)\ast z=x\ast(y\ast z)\;(x,y,z\in[0,1])\)) and symmetric (\(x*y=y*x\;(x,y\in[0,1])\)) function of two variables, nondecreasing in each variable, satisfying the boundary conditions (B) \(x*0=0,\;x*1=x\) for all \(x\in[0,1]\); in other words a commutative, order preserving semigroup operation on \([0,1]\) with identity \(1\) and null-element \(0.\) A t-norm is strict if it is continuous on \([0,1]^2\) and strictly increases in each variable on \(]0,1].\) A copula (don’t conjugate) is a function \(\ast: [0,1]^2\to [0,1]\) that satisfies the boundary conditions (B) and the inequality \[ x\ast y - u\ast y - x\ast v + u\ast v \geq 0\quad(0\leq x\leq u\leq 1,\; 0\leq y\leq v\leq 1). \]

The authors initially rely upon the following two results.

(A) Let \(J\) be a subinterval of positive length of \(\mathbb{R}\). Then the function \(\:* : J^2\to J\) is associative \((x*y)*z=x*(y*z)\; (x,y,z\in J)\) continuous and strictly increasing in each variable iff there exists a continuous and strictly monotonic function \(f: J\to \mathbb{R},\) with inverse \(f^{-1}\) such that \(x*y=f^{-1}[f(x)+f(y)] \;(x,y\in J).\) Moreover, this is possible only if \(J\) is open or half-open [cf. J. Aczél, Bull. Soc. Math. Fr. 76, 59–64 (1948; Zbl 0033.11002)].

(L) If \(*\) is a continuous and archimedean (\(x*x<x\) for \(x\in ]0,1[\)) t-norm then there exist continuous nondecreasing functions \(f,g\) such that \(x*y=g[f(x)+f(y)]\) \((x,y\in [0,1])\) [cf. C.-H. Ling, Publ. Math. 12, 189–212 (1965; Zbl 0137.26401)].

As the authors note, this book, together with that by E. P. Klement, R. Mesiar and E. Pap [Triangular norms. Dordrecht: Kluwer (2000; Zbl 0972.03002)] pretty well cover the terrain of triangular norms, copulas, and many of their applications.

In addition to mathematical generalizations and connections, the present book indicates applications among others to random variables, probability distributions, marginals, correlation, probabilistic metric spaces, fuzzy sets, fuzzy logic, theory of information wihout probability, Ulam-Hyers stability, etc.

The chapters are: 1. Introduction, 2. Representation theorems for associative functions, 3. Functional equations involving t-norms [including also simultaneous associativity, homogeneity, and distributivity; the translation equation and its generalization have been touched already in subsection 2.8, the Cauchy and Jensen equations and inequalities and several functional equations in a single variable (meaning there is only one variable in the equation), even sooner] and 4. Inequalities involving t-norms [including convexity and dominance]. There are two appendices: Examples and counterexamples, and Open problems.

We will, however, define here t-norms and copulas using the latter notation: A t-norm \(* : [0,1]^2\to[0,1]\) is an associative (\((x\ast y)\ast z=x\ast(y\ast z)\;(x,y,z\in[0,1])\)) and symmetric (\(x*y=y*x\;(x,y\in[0,1])\)) function of two variables, nondecreasing in each variable, satisfying the boundary conditions (B) \(x*0=0,\;x*1=x\) for all \(x\in[0,1]\); in other words a commutative, order preserving semigroup operation on \([0,1]\) with identity \(1\) and null-element \(0.\) A t-norm is strict if it is continuous on \([0,1]^2\) and strictly increases in each variable on \(]0,1].\) A copula (don’t conjugate) is a function \(\ast: [0,1]^2\to [0,1]\) that satisfies the boundary conditions (B) and the inequality \[ x\ast y - u\ast y - x\ast v + u\ast v \geq 0\quad(0\leq x\leq u\leq 1,\; 0\leq y\leq v\leq 1). \]

The authors initially rely upon the following two results.

(A) Let \(J\) be a subinterval of positive length of \(\mathbb{R}\). Then the function \(\:* : J^2\to J\) is associative \((x*y)*z=x*(y*z)\; (x,y,z\in J)\) continuous and strictly increasing in each variable iff there exists a continuous and strictly monotonic function \(f: J\to \mathbb{R},\) with inverse \(f^{-1}\) such that \(x*y=f^{-1}[f(x)+f(y)] \;(x,y\in J).\) Moreover, this is possible only if \(J\) is open or half-open [cf. J. Aczél, Bull. Soc. Math. Fr. 76, 59–64 (1948; Zbl 0033.11002)].

(L) If \(*\) is a continuous and archimedean (\(x*x<x\) for \(x\in ]0,1[\)) t-norm then there exist continuous nondecreasing functions \(f,g\) such that \(x*y=g[f(x)+f(y)]\) \((x,y\in [0,1])\) [cf. C.-H. Ling, Publ. Math. 12, 189–212 (1965; Zbl 0137.26401)].

As the authors note, this book, together with that by E. P. Klement, R. Mesiar and E. Pap [Triangular norms. Dordrecht: Kluwer (2000; Zbl 0972.03002)] pretty well cover the terrain of triangular norms, copulas, and many of their applications.

In addition to mathematical generalizations and connections, the present book indicates applications among others to random variables, probability distributions, marginals, correlation, probabilistic metric spaces, fuzzy sets, fuzzy logic, theory of information wihout probability, Ulam-Hyers stability, etc.

The chapters are: 1. Introduction, 2. Representation theorems for associative functions, 3. Functional equations involving t-norms [including also simultaneous associativity, homogeneity, and distributivity; the translation equation and its generalization have been touched already in subsection 2.8, the Cauchy and Jensen equations and inequalities and several functional equations in a single variable (meaning there is only one variable in the equation), even sooner] and 4. Inequalities involving t-norms [including convexity and dominance]. There are two appendices: Examples and counterexamples, and Open problems.

Reviewer: János Aczél (Waterloo/Ontario)

### MSC:

39B22 | Functional equations for real functions |

03E72 | Theory of fuzzy sets, etc. |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

46N30 | Applications of functional analysis in probability theory and statistics |

46S40 | Fuzzy functional analysis |

60A10 | Probabilistic measure theory |