Smooth t-subnorms on finite scales
Smoothness of functions on a finite scale \(L= \{0,1,\dots, n\}\) means simply the \(1\)-Lipschitzianity (with respect to the \(L_1\)-norm). The authors study smooth triangular subnorms (associative monotone commutative binary operations on \(L\) bounded by both input elements) and smooth weak boundary triangular norms (smooth triangular subnorms with \(n\) as an idempotent element). Smooth bwt-norms are shown to coincide with smooth triangular norm, i.e., \(n\) is their neutral element, which were completely characterized in [\textit{G. Mayor} and \textit{J. Torrens}, ``Triangular norms on discrete settings'', in: E. P. Klement (ed.) et al., Logical, algebraic, analytic and probabilistic aspects of triangular norms. Selected papers from the 24th Linz seminar on fuzzy set theory, Linz, Austria, 2003. Amsterdam: Elsevier. 189--230 (2005; Zbl 1079.03012)]; see also [\textit{L. Godo} and \textit{V. Torra}, ``On aggregation operators for ordinal qualitative information'', IEEE Trans. Fuzzy Syst. 8, No. 2, 143--154 (2000)]. Smooth triangular subnorms are shown to have a similar structure up to the last summand in their ordinal sum representation. This last summand should be a smooth Archimedean triangular subnorm on a scale \(K= \{0,\dots, k\}\) (cf. a similar situation in the continuous case [\textit{A. Mesiarová}, ``Continuous triangular subnorms'', Fuzzy Sets Syst. 142, No. 1, 75--83 (2004; Zbl 1043.03018)]). Although a complete characterization of smooth Archimedean triangular subnorms is still missing (thus we have an interesting open problem!), the paper gives some partial solutions and several families of such t-subnorms.
scientific article; zbMATH DE number 5882967
scientific article