A conditionally cancellative left-continuous t-norm is not necessarily continuous

In this paper, we propose a new method for constructing non-continuous t-norms from a given t-norm. Using this method, we construct a family of conditionally cancellative left-continuous t-norms which are not continuous. Thus we answer an open question which is collected by Klement et al. [Problems on triangular norms and related operators, Fuzzy Sets and Systems 145 (2004) 471–479].