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This paper mainly focuses on new methods of constructing triangular norms as well as triangular conorms on a bounded lattice. At first, we propose the definition of a partition element and show that the definition is well-defined. Then, we construct a t-norm (t-conorm) on a bounded lattice with the help of an interior (a closure) operator and a given t-subnorm (t-subconorm). For the sake of clarity, we give some interpretive examples of the new methods and show that the conditions restricting the elements are indispensable in the construction process. Moreover, we discuss the relationships among the construction methods proposed in this paper and the methods that already exist. Finally, we generalize the new construction methods to modified ordinal sum methods by iteration. [ABSTRACT FROM AUTHOR]