Two examples of the constructions of non-continuous t-norms

In this contribution we present some interesting constructions of non-continuous triangular norms. The first approach is based on strictly increasing sequences of natural numbers. An associative commutative monotone and bounded by minimum binary operation on these sequences induces a t-norm. The corresponding t-norm is left continuous and therefore it is applicable in the fuzzy logic. Several other interesting properties of this t-norm are investigated, including its residual implicator. Second approach uses idea of multiplicative generator φ of a triangular norm is a special monotone function φ : [0, 1] → [0, 1] with fixed point 1 and φ(0) < 1. The corresponding t-norm T is defined by means of φ as follows: T ∗(x, y) = φ(φ(x).φ(y)), where φ : [0, 1] → [0, 1] is a so-called pseudo-inverse of φ. If strictly increasing function φ is left continuous, but non-continuous, then associativity of induces operator is violated, [6]. However, then the operation