  The ordinal sum construction provides a very effective way to generate a new
triangular norm on the real unit interval from existing ones. One of the most
prominent theorems concerning the ordinal sum of triangular norms on the real
unit interval states that a triangular norm is continuous if and only if it is
uniquely representable as an ordinal sum of continuous Archimedean triangular
norms. However, the ordinal sum of triangular norms on subintervals of a
bounded lattice is not always a triangular norm (even if only one summand is
involved), if one just extends the ordinal sum construction to a bounded
lattice in a na\"{\i}ve way. In the present paper, appropriately dealing with
those elements that are incomparable with the endpoints of the given
subintervals, we propose an alternative definition of ordinal sum of countably
many (finite or countably infinite) triangular norms on subintervals of a
complete lattice, where the endpoints of the subintervals constitute a chain.
The completeness requirement for the lattice is not needed when considering
finitely many triangular norms. The newly proposed ordinal sum is shown to be
always a triangular norm. Several illustrative examples are given.
