On ordinal sums of triangular norms on bounded lattices

Incomparability with respect to the triangular order

“…Some sets were determined which, under some special conditions, are lattices with respect to T . For more details on t-norms on bounded lattices, we refer to [3,9,10,11,13,15,16,17,19].…”

Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices

“…the well-known commutative po-monoid operations in Birkhoff's terminology [2] or the ordinal sum of t-norms on bounded lattices using Clifford's definition [5], which is studied in [15,16]. These papers present conditions for ensuring that, given a family of pairwise disjoint subintervals, the ordinal sum of any family of t-norms on these subintervals is always a t-norm (which is not true in general).…”

“…As a result, a straightforward tool to verify whether the results of [15,16] can be applied will be presented, in order to construct t-norms from ordinal sums of t-norms or to recognize the structure of the bounded lattices considered.…”

On the structure of left-continuous t-norms that have a continuous contour line

“…Various construction methods have been proposed for creating new t-norms (see e.g. [2,13,18,20,27]). Most of these methods start from a known t-norm on which a bunch of operations such as rotations, annihilations and rescalings is performed.…”