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

Some methods to obtain t-norms and t-conorms on bounded lattices

“…Based on the constructions of t-norms and t-conorms on the unit interval [0, 1], the study of the constructions of t-norms and t-conorms defined on bounded lattices has recently become important. In the papers [13,14,10,17,24,25], t-norms and t-conorms defined on bounded lattices were investigated and several methods for constructing these operators were introduced. Furthermore, by using the existence of a priori given t-norm and t-conorm, some ordinal sum constructions for t-norms and t-conorms on bounded lattices were presented in [10,17,24,25].…”

“…In the papers [13,14,10,17,24,25], t-norms and t-conorms defined on bounded lattices were investigated and several methods for constructing these operators were introduced. Furthermore, by using the existence of a priori given t-norm and t-conorm, some ordinal sum constructions for t-norms and t-conorms on bounded lattices were presented in [10,17,24,25]. However, in [24,25], several necessary and sufficient conditions are required in order to ensure that an ordinal sum of any t-norms (tconorm) is a t-norm (t-conorm).…”

“…Furthermore, by using the existence of a priori given t-norm and t-conorm, some ordinal sum constructions for t-norms and t-conorms on bounded lattices were presented in [10,17,24,25]. However, in [24,25], several necessary and sufficient conditions are required in order to ensure that an ordinal sum of any t-norms (tconorm) is a t-norm (t-conorm). We aim to enhance these constructions by introducing the different construction methods for t-norms and t-conorms valid on any bounded lattice.…”

“…In this construction, some necessary and sufficient conditions are presented for the ordinal sum on a bounded lattice of arbitrary t-norms (t-conorms) to yield a t-norm (t-conorm). The other investigations of ordinal sum constructions for obtaining t-norms and t-conorms on bounded lattices can be found in [24], where also some additional conditions are required in order to ensure that an ordinal sum of arbitrary t-norms (tconorms) is a t-norm (t-conorm). Uninorm-like operations as a generalization of t-norms and t-conorms with the underlying operations given by ordinal sums were studied in [15].…”

A binary operation-based representation of a lattice

“…Furthermore, it is not surprising that binary operations with specific properties appear in various theoretical and application fields. For instance, for aggregation functions (as specific binary operations) on bounded lattices and their wide use in various fields of applied sciences, including, computer and information sciences, economics, and social sciences (see, e. g. [12,13,14,15,17,21,22]). Also, they play an important role (as generalization of the basic connectives between fuzzy sets) in theories of fuzzy sets and fuzzy logic [2].…”