On the construction of boundary weak triangular norms through additive generators

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)=xT(x,1)=x by the weaker one T(x,1)≤xT(x,1)≤x for x∈[0,1)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,∞]f:[0,1]→[0,∞] which generate proper boundary weak triangular norms are completely characterized. Several examples are included.