Construction of uninorms on bounded lattices

Uninorms Having an Identity and an Annihilator on Bounded Lattices

“…Furthermore, a uninorm on some bounded lattices which an identity g is incomparable with an annihilator need not always exist was proved. The uninorms on bounded lattices were also discussed by several researchers in other studies (Bodjanova and Kalina, 2014;Çaylı and Karaçal, 2017;Aşıcı, 2018;Çaylı and Drygaś, 2018).…”

New construction approaches of uninorms on bounded lattices

“…Bodjanova and Kalina (2019) described the classes of uninorms derived from both t-norms and t-conorms on bounded lattices. Some construction methods for yielding new classes of uninorms on bounded lattices were also proposed in the studies (Aşıcı and Mesiar 2020b;Bodjanova and Kalina 2018;Çaylı 2018, 2019, 2020Çaylı and Karaçal 2017;Çaylı, Karaçal, and Mesiar 2019;Dan and Hu 2020;Dan, Hu, and Qiao 2019;Ouyang and Zhang 2020;Xie and Li 2020). In this article, based on the presence of a t-norm T e on [0, e] 2 and a tconorm S e on [e, 1] 2 , we introduce some new construction methods to obtain uninorms with the identity e ∈ L \ {0, 1} by use of closure (interior) operators on a bounded lattice L. As a by-product, we obtain two families of idempotent uninorms on bounded lattices if we take that T e = T ∧ on [0, e] 2 and S e = S ∨ on [e, 1] 2 .…”

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

“…An operation T : L 2 → L is called a t-norm on L if it is commutative, associative, increasing with respect to both variables and has the neutral element 1 such that T (x, 1) = x, for all x ∈ L. Definition 2.5. (Aşıcı and Karaçal [1], Aşıcı [2,3], Ç aylı and Karaçal [8]) Let (L, ≤ , 0, 1) be a bounded lattice. An operation S : L 2 → L is called a t-conorm on L if it is commutative, associative, increasing with respect to both variables and has the neutral element 0 such that S (x, 0) = x, for all x ∈ L.…”

“…In order to avoid this problem, some modified versions of the above mentioned ordinal sum construction were proposed in [10,17]. Considering an arbitrary bounded lattice L, for any element a ∈ L\{0, 1}, based on a t-norm V acting on the subinterval [a, 1], the constructions given by the formulas (5) and 7, respectively, in Theorems 2.13 and 2.14 yield a t-norm on L. Similarly, based on a t-norm W acting on the subinterval [0, a], the constructions given by the formulas (6) and (8), respectively, in Theorems 2.13 and 2.14 yield a t-conorm on L.…”