Continuous completions of triangular norms known on a subregion of the unit interval

The continuous extensions of a t-norm know only on some subregion of the unit interval to the whole unit interval are studied. The conditions which a t-norm T1 should satisfy on [a,b]2 necessary for the existence of a continuous t-norm T2 that coincides with T1 on [a,b]2 are given. Construction of such a continuous t-norm T2 is described.