Beyond left-continuity: t-norms on the fuzzy unit interval induced by pairs of t-norms

We define a binary operation $T_1\star T_2$ on the fuzzy unit interval \I by extending the natural action of $T_1$ and $T_2$ on step functions. When both $T_1$ and $T_2$ are left-continuous, this operation coincides with the tensor product of the corresponding quantales and therefore yields a t-norm on \I. Without left-continuity, the situation becomes more subtle. We prove that if at least one of the two t-norms is continuous, then $T_1\star T_2$ is still a t-norm on \I. We also show by counterexample that continuity cannot simply be dropped. The operation \DN fails to be associative. On the other hand, $T_D\star T_D$ is a t-norm despite $T_D$ being neither continuous nor left-continuous, indicating that continuity is not a necessary condition.