On the construction of boundary weak triangular norms through additive generators

Abstract

Recently, M. Urbański and J. Wa̧sowski [Fuzzy arithmetic based on boundary weak t-norms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 13 (2005) 27–37] extended the triangular norm operators to a new class of operators called the boundary weak triangular norms. This extension is based on the replacement of the condition T ( x , 1 ) = x by the weaker one T ( x , 1 ) ≤ x for x ∈ [ 0 , 1 ) . In this paper we discuss the construction of the boundary weak triangular norms based on their additive generators. Continuous decreasing functions f : [ 0 , 1 ] → [ 0 , ∞ ] which generate proper boundary weak triangular norms are completely characterized. Several examples are included.