How to costruct left-continuous triangular norms

Triangular norms (t-norms for short) play a crucial role in several fields of mathematics and AI. For an exhaustive overview on t-norms we refer to [20]. Recently an increasing interest of left-continuous t-norm based theories can be observed (see e.g. [3, 5, 6, 7, 18]). In this paper we discuss in detail the presently existing construction methods which result in left-continuous triangular norms. The methods are (together with their sources): annihilation [12, 2], ordinal sum of t-subnorms [11, 9], rotation contruction [14, 8], rotation-annihilation construction [16], embedding method [17, 6]. An infinite number of left-continuous triangular norms can be generated with these constructions (and with their combinations), which provides a tremendously wide spectrum of choice for e.g. logical and set theoretical connectives in non-classical logic and in fuzzy theory.