Fuzzy Sets and Systems 40 (1991) 431-450 431 North-Holland Theory of T-norms and fuzzy inference methods M.M. Gupta and J. Qi Intelligent Systems Research Laboratory, College of Engineering, University of Saskatchewan, Saskatoon, Sask., Canada S7N OWO Received November 1989 Revised July 1990 Abstract: In this paper, the theory of T-norm and T-conorrn is reviewed and the T-norm, T-conorm and negation function are defined as a set of T-operators. Some typical T-operators and their mathematical properties are presented. Finally, the T-operators are extended to the conventional fuzzy reasoning methods which are based on the MIN and MAX operators. This extended fuzzy reasoning provides both a general and a flexible method for the design of fuzzy logic controllers and, more generally, for the modelling of any decision-making process. Keywords: T-norms; T-eonorms; T-operators; fuzzy inference; fuzzy logic controller. 1. Introduction The triangular norm (T-norm) and the triangular conorm (T-conorm) origin- ated from the studies of probabilistic metric spaces [1, 2] in which triangular inequalities were extended using the theory of T-norm and T-conorm. Later, H6hle [3], Alsina et al. [4], etc. introduced the T-norm and the T-conorm into fuzzy set theory and suggested that the T-norm and the T-conorm be used for the intersection and union of fuzzy sets. Since then, many other researchers have presented various types of T-operators for the same purpose [5, 6, 7] and even proposed some methods to generate the variations of these operators [8] which are given in the Appendix. Zadeh's conventional T-operators, MIN and MAX, have been used in almost every design of fuzzy logic controllers and even in the modelling of other decision-making processes. However, some theoretical and experimental studies seem to indicate that other types of T-operators may work better in some situations, especially in the context of decision-making processes. For example, the product operator may be preferred to the MtN operator [9]. On the other hand, when choosing a set of T-operators for a given decision-making process, one has to consider their properties, the accuracy of the model, their simplicity, computer and hardware implementation, etc. For these and other reasons, it is of interest to use other sets of T-operators in the modelling of decision-making processes, so that multiple options are available for selecting T-operators that may be better suited for given problems. In this paper we will give a detailed exposition of the theory of T-operators, the various methods of their generations, and possible applications in fuzzy reasoning processes. 0165-0114/91/$03.50 (~ 1991--Elsevier Science Publishers B.V. (North-Holland)