Ordinal sums of triangular norms on a bounded lattice

Abstract

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 naïve 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.

Triangular norms on bounded trellises

“…Notwithstanding the many achievements mentioned above, all situated in the comfortable setting of real analysis, it took until the 90s for the study of t-norms on more general structures to take off [12,13], in particular on partially ordered sets (posets, for short) and lattices, thus being related to the study of partially ordered semigroups [21]. Not surprisingly, research in this direction has been a lot slower, initially being focused on construction methods [1,2,7,25,28], and only recently more profound characterizations in terms of ordinal sums [16,33,34].…”

On the constructions of t-norms and t-conorms on some special classes of bounded lattices

“…In 2015, a modification of ordinal sums of t-norms and t-conorms resulting to a t-norms and t-conorms on an arbitrary bounded lattice was shown by Ertugrul, Karaçal, Mesiar [16]. Further modifications were proposed by Ç aylı [11,13], Aşıcı, Mesiar [3,4], Ouyang, Zhang, Baets [23] and Dan, Hu, Qiao [14]. In 2020, a new ordinal sum construction of t-norms and t-conorms on bounded lattices based on interior and closure operators was proposed by Dvoȓák, Holcapek [15].…”