The smallest semicopula-based universal integrals I: Properties and characterizations

In this article we provide a detailed study of basic properties of the smallest universal integral IS with S being the underlying semicopula and review some of the recent developments in this direction. The class of integrals under study is also known under the name seminormed integrals and includes the well-known Sugeno and Shilkret integrals as special cases. We present some representations and equivalent formulations for IS. Further we discuss basic properties such as translatability, linearity, homogeneity, and maxitivity. In particular, we provide the characterizations of the two prominent integrals of Sugeno and Shilkret. We also raise and discuss some open questions.