The encounter, in the late seventies, between the theory of triangular
norms, issuing frorn stochastic geornetry, especially the works of
Menger, Schweizer and Sklar, on the one band, and the theory of fuzzy
sets due to Zadeh, 10n the other band has been very fruitful. Triangular
norms have proved to be ready-rnade mathematical rnodels of fuzzy set
intersections and have shed light on the algebraic foundations of fuzzy
sets. One basic idea behind the study of triangular norms is to solve
functional equations that stern frorn prescribed axioms describing
algebraic properties such as associativity. Alternative operations such
as rneans have been characterized in a similar way by Kolmogorov, for
instance, and the rnethods for solving functional equations are now weil
established thanks to the efforts of Aczel, among others. One can say
without overstaternent that the introduction of triangular norms in
fuzzy sets has strongly influenced further developrnents in fuzzy set
theory, and has significantly contributed to its better acceptance in
pure and applied rnathematics circles. The book by Fodor and Roubens
systematically exploits the benefits of this encounter in the- analysis
of fuzzy relations. The authors apply functional equation rnethods to
notions such as equivalence relations, and various kinds of orderings,
for the purpose of preference rnodelling. Centtal to this book is the
rnultivalued extension of the well-known result claiming that any
relation expressing weak preference can be separated into three
cornponents respectively describing strict preference, indifference and
incomparability.
