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.

This paper is mainly devoted to investigating the migrativity equations involving nullnorms and 2-uninorms. Depending on whether the absorbing elements of 2-uninorms and unllnorms are same or not, all solutions of the migrativity equations for all possible combinations of the three defined subclasses of 2-uninorms and nullnorms are analyzed and characterized respectively. And for such equations, there are new solutions which extend the known ones about the migrativity for uninorms and nullnorms.

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.

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.

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.