Uninorms were introduced by Yager and Rybalov as a generalization of both t-norms and t-conorms. They have been studied extensively both as aggregation functions and as logical connectives. Describing the full structure of the entire class of uninorms remains challenging, and thus several classes of uninorms have been put forward. In 2015, Mas et al. compiled the existing classes of uninorms in two primary frameworks: the real unit interval [ 0 , 1 ] and the discrete setting. However, in recent years, theoretical research on the characterization of the classes of uninorms on the real unit interval has made significant progress, especially for the class of uninorms with continuous underlying functions, the class of uninorms that are internal on the boundary, and the class of uninorms that are internal on non-trivial cuts. This review presents a careful selection of results that shed light on the structure of the members of the most prominent classes of uninorms on the real unit interval. The selected results describe the structure of uninorms in a way similar to the well-known structural description of continuous t-norms credited to Ling, based on a result by Mostert and Shields. To ensure the comprehensive and self-contained nature of our presentation, findings from before 2015 are also briefly presented, thereby also facilitating the verification of subsequent discoveries. Several significant results are omitted due to the limited space; however, the major part of the quintessential references is provided for the interested reader.

This article focuses on distributivity and conditional distributivity for uninorms over a given continuous t-norm, closely related to the open problem recalled by Klement in the Linz2000 closing session. This problem has been solved for the well-known classes of uninorms except the class of uninorms with the continuous underlying operators. In this article, the complete characterizations of uninorms with continuous underlying operators (conditionally) distributive over a given continuous t-norm are presented. From the obtained results, we deduce that conditional distributivity is equivalent to distributivity for a pair of such operators.

Distributivity between two operations is a property posed many years ago — that is especially interesting in the framework of logical connectives because of its applications to fuzzy logic and approximate reasoning as their applications. Since semi-uninorms have been used in these topics, the study of the distributivity between two semi-uninorms becomes of particular interest that calls for thorough studies. The distributivity between two semi-uninorms, which are non-commutative and non-associative uninorms, has been developed only in the cases when both semi-uninorms are examples of very special classes of semi-uninorms. On the other hand, in general, the distributivity does not rely on the commutativity and associativity. The objective of this work is twofold. The first one is to show new solutions to distributivity equations for semi-uninorms. The second one is to check whether the results concerning the distributivity between two uninorms are valid for semi-uninorms. We investigate the distributivity involving two semi-uninorms when only one semi-uninrom lies in the most studied classes of semi-uninorms, achieving the above two objectives simultaneously.

Recently, the α-migrativity has been discussed in families of certain operations (e.g., triangular norms, conorms, uninorms, and nullnorms). In this paper, the migrative property for semi-t-operators is studied. All solutions of the migrativity equation for all possible combinations of semi-t-operators are analyzed and characterized.

It is known that bisymmetry generalizes the simultaneous commutativity and associativity in the framework of the unit interval. In this work, we will completely characterize two classes of bisymmetric aggregation operators: one with a neutral element and the other with the vertical and horizontal sections of the idempotent elements being smooth on a finite chain, but not necessarily smooth and commutative. Thus, the previous results, based on the smoothness that is known as a very restrictive condition, are improved. For example, there is only one smooth Archimedean t-norm on a finite chain. In this paper, the discrete bisymmetric aggregation operators are explored without the limit of the smoothness. As a by-product, it is deduced that for smooth aggregation operators on a finite chain, the bisymmetry is equivalent to the commutativity and associativity, which improves the conclusion obtained by Mas et al. that associativity and bisymmetry are equivalent for commutative smooth aggregation operators on a finite chain.

Recently, Su and Liu have introduced a new class of fuzzy implications, called ordinal sum implications, and discussed some of their desirable properties, such as neutrality property, consequent boundary, exchange principle, etc. In this paper, we explore the class of ordinal sum implications with respect to distributivity. Necessary and sufficient conditions, under which ordinal sum implications are distributive over t-norms and t-conorms are given.

In this paper we generalize (U,N)-implications. First we introduce (US, N)-operation by replacing the uninorm U with a semi-uninorm US in the definition of a (U, N)-operation. Then the necessary and sufficient conditions that a (US, N)-operation is a implication are given out. Furthermore, we characterize the (US, N)-implications. Last we generalize the definition of rotation invariant and investigate the intersection between the residual implications and (US,N)-implications derived from some semi-uninorms with the generalized rotation invariant.

In this paper, we firstly introduce two new classes of fuzzy implications generated from one-variable functions, called [Formula: see text]- and [Formula: see text]-implications, respectively. Then we give a series of necessary and sufficient conditions that these implications satisfy: left neutrality property, identity principle, ordering principle, law of contraposition, modus ponens and modus tollens, respectively. We also discuss the relations between [Formula: see text]- implication ([Formula: see text]-implications, respectively) and other known classes of fuzzy implications.

The α-migrativity is an important property of a binary operation, and many authors discussed this property for various operators. In this paper, the migrative property for nullnorms is investigated. All solutions of the migrativity equation for all possible combinations of nullnorms are analyzed and characterized.