Solution of an open problem on pseudo-Archimedean t-norms
Let \((P,\wedge,\vee,\bot,\top)\) be a bounded lattice. A t-norm \(T\) on the lattice is any commutative and associative binary operation on \(P\) which is non-decreasing with respect to the lattice order and has the unit element \(\top\). For the special case when the lattice is the chain \([0,1]\) it is known that every t-norm with the pseudo-Archimedean property which is not cancellative has a nilpotent element. In the paper under review the authors show that this is no longer true for t-norms on general bounded lattices. Namely, they construct two examples of a bounded lattice and for each one of them a pseudo-Archimedean t-norm which is not cancellative and has no nilpotent elements. This work was motivated by an open question posed by \textit{B. De Baets} and \textit{R. Mesiar} [Fuzzy Sets Syst. 104, No.~1, 61--75 (1999; Zbl 0935.03060)].
scientific article; zbMATH DE number 5531313