Associativity of Triangular Norms in Light of Web Geometry

The aim of this paper is to promote web geometry and, especially, the Reidemeister closure condition as a powerful and intuitive tool characterizing associativity of the Archimedean triangular norms. In order to demonstrate its possible applications, we provide the full solution to the problem of convex combinations of nilpotent triangular norms. Keywords— Archimedean triangular norm, web geometry, Reidemeister closure condition. 1 Triangular norms The notion of triangular norm was originally introduced within the framework of probabilistic metric spaces [12]. Since then, triangular norms have found diverse applications in the theory of fuzzy sets, fuzzy decision making, in models of certain many-valued logics or in multivariate statistical analysis; for a reference see the books by Alsina, Frank, and Schweizer [5] and by Klement, Mesiar, and Pap [8]. A conjunctor is a function K : [0, 1] → [0, 1] which is nondecreasing in both arguments, commutative, and which satisfies the boundary condition T (x, 1) = x for all x ∈ [0, 1]. A triangular norm (shortly a t-norm, usually denoted by T ) is a conjunctor which satisfies the associativity equation T (T (x, y), z) = T (x, T (y, z)) for all x, y, z ∈ [0, 1]. This paper deals mainly with Archimedean t-norms. Let us recall that for every t-norm T , a number n ∈ N∪{0}, and x ∈ [0, 1] a natural power of x, denoted (x) T , is defined by: (x) T = { 1 if n = 0 , T ( x, (x) T ) if n > 0 . (1) A t-norm T is said to be Archimedean if and only if for every pair x, y ∈ ]0, 1[, x < y, there exists a natural number n ∈ N such that (y) T < x. A t-norm which is continuous and strictly increasing on the half-open square ]0, 1] is said to be strict. A continuous t-norm T is called nilpotent if and only if for every x ∈ ]0, 1[ there exists a natural number n ∈ N such that (x) T = 0. A prototypical example of a strict and a nilpotent t-norm is the product t-norm, TP(x, y) = x · y, and the Łukasiewicz t-norm TL(x, y) = max{x + y − 1, 0}, Figure 1: Example of a complete 3-web. respectively. It is possible to show that a continuous Archimedean t-norm is either strict or nilpotent. A t-norm T is said to be cancellative if it satisfies T (a, b) = T (a, c) ⇒ b = c for every a, b, c ∈ [0, 1], a = 0. A t-norm T is said to be weakly cancellative [9] if it satisfies T (a, b) = T (a, c) ⇒ b = c for every a, b, c ∈ [0, 1] with T (a, b) = 0 and T (a, c) = 0. Every cancellative t-norm is weakly cancellative. Under the assumption of continuity, the set of cancelative t-norms and the set of strict t-norms coincide. Under the same assumption, the set of weakly cancelative t-norms coincides with the set of Archimedean t-norms. 2 Web geometry and local loops In this section, web geometry is explained as a tool allowing to visualize algebraic identities. In particular, it is shown that it visualizes associativity of Archimedean t-norms. A detailed introduction to the subject is given in the monograph by Blashke and Bol [6]. Also the collection of papers by Aczél, Akivis, and Goldberg [1, 2, 3] can serve as an (English) introductory text. Definition 2.1 A groupoid (or a magma) is an algebraic structure G = (G, ◦) on a set G where ◦ : G × G → G is a binary operation. A quasigroup is a groupoid in which the ISBN: 978-989-95079-6-8 IFSA-EUSFLAT 2009