A state-of-the-art survey of the most prominent classes of uninorms on the unit interval

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.