Ordinal sums of triangular norms on a bounded lattice

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