On the continuity points of left-continuous t-norms
The authors show that each left-continuous t-norm can be obtained as the completion of a continuous function on some dense subset of \([0,1]^2\). As a by-product, they prove that the set of discontinuity points of a left-continuous t-norm is a first-category set of measure zero. Further, they show that not all left-continuous t-norms are isomorphic to the completion of some continuous t-norm on~\(Q\cap[0,1]\) (the set of all rational numbers from the unit interval). However, they prove that at least all weakly cancellative ones possess this property. Finally, they characterize those continuous t-norms on \(Q\cap[0,1]\) whose completion is continuous.
scientific article; zbMATH DE number 2051464
scientific article