T-norms and t-conorms continuous around diagonals

Triangular norms and conorms with continuous diagonals are discussed. In literature we can find examples of non-continuous t-norms with continuous diagonals, however, their deeper study is still missing. In this paper we introduce a sufficient condition under which a t-norm with continuous diagonal is continuous. Moreover, we show that an Archimedean t-norm continuous on the boundary is continuous. Several illustrative examples are also included.