A characterization of the classes Umin and Umax of uninorms on a bounded lattice

Idempotent uninorms on a complete chain

“…The restriction of a uninorm U to [0, e] 2 is a t-norm on [0, e], while its restriction to [e, 1] 2 is a t-conorm on [e, 1]. Recently, we introduced two classes of uninorms on a general bounded lattice by generalizing the two well-known classes U min and U max of uninorms on the real unit interval and characterized the structure of their members [18].…”

“…Let g be a decreasing unary function on L. Then g has property ( S) if and only if it satisfies g(g(x)) ≥ x , for all x ∈ [0, g(0)] (17) and g(x) = 0 , for all x > g(0). (18) Proof. Suppose that g satisfies ( S).…”

“…For any x > g(0), since g also has property (S) (see Proposition 5.5), it follows that g(x) = 0. Therefore, g also satisfies (18). Suppose that g satisfies ( 17) and (18).…”

“…In 2015, Karaçal and Mesiar [13] straightforwardly generalized the notion of a uninorm from the real unit interval to the more general setting of bounded lattices and identified the smallest and greatest uninorms with a given neutral element on a given bounded lattice. Recently, we [18] generalized the classes U min and U max of uninorms on the real unit interval to bounded lattices and characterized the structure of their members. Çaylı et al [2] demonstrated the existence of idempotent uninorms with an arbitrary neutral element on an arbitrary bounded lattice.…”

New structures for uninorms on bounded lattices

“…Recently, because of "For general systems, where we cannot always expect real (or comparable) data, this extension of the underlying career together with its structure is significant..." [7], the researchers widely study uninorms on the bounded lattices instead of the unit interval [0, 1]. About the methods for the construction of uninorms, they mainly focus on t-norms (t-conorms) [1,[3][4][5][6]8,9,11,19,23], t-subnorms ( t-subconorms) [15,18,25], closure operators ( interior operators) [7,16,21,26] and additive generators [17].…”