Solution to an open problem: a characterization of conditionally cancellative t-subnorms
The author proves that for any \(t\)-subnorm \(M\!:[0,1]^2\rightarrow [0,1]\), having the residual operation \(I(x,y)=\sup\{t\in [0,1] \mid M(x,t)\leq y\}\), \(x,y\in [0,1]\), the following conditions are equivalent: (i) \(I(x,M(x,y))=\max(n(x),y)\), \(x,y\in [0,1]\), with \(n(x)=I(x,0)\) for all \(x\in [0,1]\) and (ii) \(M\) is a conditionally cancelative \(t\)-subnorm. This is the solution to problem 11 of U. Höhle [\textit{E. P. Klement, R. Mesiar} and \textit{E. Pap}, Fuzzy Sets Syst. 145, No. 3, 471--479 (2004; Zbl 1050.03019)]. Finally, this result leads to the conclusion of the non-existence of a conditionally cancelative \(t\)-subnorm whose natural negation is involutive.
scientific article; zbMATH DE number 6116949
scientific article