A new extension of a triangular norm on a subinterval $[0,\alpha]$ via an interior operator to the underlying entire bounded lattice

As a proper generalization of the ordinal sum t-norm construction on bounded lattices proposed in [E. A\c{s}{\i}c{\i}, R. Mesiar, New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice, International Journal of General Systems, {\bf 49}(2) (2020), 143-160], the present paper studies a new extension of a triangular norm on a subinterval $[0,\alpha]$ via an interior operator to the underlying entire bounded lattice, where the necessary and sufficient conditions under which the constructed operation is again a t-norm are given. By comparing the graphic structures of two t-norms on a common bounded lattice which are constructed in different ways, it is shown that the new method in this paper is essentially different from the ones existing in the literature. As an end, this new construction is generalized to construct ordinal sums of finitely many t-norms by recursion on bounded lattices. The dual results for ordinal sum construction of t-conorms via closure operators on bounded lattices are also presented.