Commun. Korean Math. Soc. 32 (2017), No. 2, pp. 353–359 https://doi.org/10.4134/CKMS.c160089 pISSN: 1225-1763 / eISSN: 2234-3024 ON A CLASS OF GENERALIZED TRIANGULAR NORMS Iqbal Jebril and Mustapha Ra¨ ıssouli Abstract. Starting from a t-norm T , it is possible to construct a class of new t-norms, so-called T -generalized t-norm. The purpose of this paper is to describe some properties of this class of generalized t-norms. An algebraic structure as well as a binary relation among t-norms are also investigated. Some open problems are discussed as well. 1. Introduction Triangular norms are operations which represent conjunctions in fuzzy logic. They were also used in the context of probabilistic metric spaces as a special kind of associative laws defined on the unit interval, see [4]. Geometrically, a triangular norm may be visualized as a surface over the unit square that contains the skew quadrilateral whose vertices are (0, 0, 0), (1, 0, 0), (1, 1, 1) and (0, 1, 0). For more detail about geometrical illustration of triangular norms, see [4] for instance. A map T : [0, 1] × [0, 1] −→ [0, 1] is called a triangular norm (in short a t-norm), if the following requirements are satisfied, see [1, 2, 3], (i) T (x, y)= T (y,x) for all x, y ∈ [0, 1], (T is commutative) (ii) T (x 1 ,y 1 ) ≤ T (x 2 ,y 2 ) whenever x 1 ≤ x 2 and y 1 ≤ y 2 ,(T is increasing) (iii) T ( x, T (y,z) ) = T ( T (x, y),z ) for all x, y, z ∈ [0, 1], (T is associative) (iv) T (x, 1) = x for all x ∈ [0, 1], (T has 1 as identity). As it is well known, combining (i), (ii) and (iv) we can see that every t-norm T satisfies T (x, y) ≤ min(x, y) for all x, y ∈ [0, 1]. It follows that T (x, x) ≤ x for every x ∈ [0, 1], with T (0, 0) = 0 and T (1, 1) = 1. Typical examples of t-norms include the following: (1) M (x, y) = min(x, y); Π(x, y)= xy; L(x, y) = max(x + y − 1, 0); (2) W (x, 1) = x, W (1,y)= y, W (x, y) = 0 if x, y ∈ [0, 1); (3) N (x, y) = min(x, y) if x + y ≥ 1, N (x, y) = 0 if x + y< 1; (3) H(0, 0) = 0,H(x, y)= xy x+y−xy if (x, y) = (0, 0); Received April 19, 2016. 2010 Mathematics Subject Classification. 08A05. Key words and phrases. t-norm, generalized t-norm, power t-norm, group of t-norms, equivalence relation between t-norms. c 2017 Korean Mathematical Society 353