Distributivity between semi-t-operators and semi-nullnorms

Distributivity equations in the class of semi-t-operators

“…The characterization of such binary operations is interesting from a theoretical point of view, but also in terms of their applications because they have proved useful in several fields, such as fuzzy logic frameworks [12], expert systems [15], neural networks [15], and fuzzy quantifiers [12]. This research is also the complement of the results of our previous study [9].…”

“…We distinguish 24 different cases, which depend on the order between the elements a, b of operation F and the elements c, d of operation G. If c = d, then we obtain the cases considered in [9], and if a = b as well, then we obtain the cases considered in [19].…”

“…A new topic of study in this area is mainly concerned with the distributivity between triangular norms and triangular conorms ( [10], p. 17). Recently, many studies have addressed the solutions of distributivity equations for aggregation functions [3], fuzzy implications [2], uninorms and nullnorms [16,21], and semi-nullnorms [9], which are generalizations of triangular norms and conorms.…”

“…Next, we characterize the solutions to distributivity equations from described families (Section 4). Finally, our results are applied to semi-nullnorms and nullnorms, which can be compared with the results reported by [16], [6], and [9] (Section 5).…”

Semi-t-operators on a finite totally ordered set

“…It is reasoned in [17] that exactly these properties are adequated for operators that have to be used in fuzzy preorders. Recently, Drygaś generalized nullnorms and t-operators and introduced semi-t-operators [4] by eliminating commutativity from the axiom of t-operators.…”