The state-of-art of the generalizations of the Choquet integral: From aggregation and pre-aggregation to ordered directionally monotone functions

In 2013, Barrenechea et al. used the Choquet integral as an aggregation function in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems. After that, starting from 2016, new aggregation-like functions generalizing the Choquet integral have appeared in the literature, in particular in the works by Lucca et al. Those generalizations of the Choquet integral, namely C-T-integrals (by t-norm T), C-F-integrals(by a fusion function F satisfying some specific properties), CC-integrals (by a copula C), C-F1F2-integrals (by a pair of fusion functions (F-1, F-2) under some specific constraints) and their generalization gC(F)(1F2)-integrals, achieved excellent results in classification problems. The works by Lucca et al. showed that the aggregation task in a FRM may be performed by either aggregation, pre-aggregation or just ordered directional monotonic functions satisfying some boundary conditions, that is, it is not necessary to have an aggregation function to obtain competitive results in classification. The aim of this paper is to present and discuss such generalizations of the Choquet integral, offering a general panorama of the state of the art, showing the relations and intersections among such five classes of generalizations. First, we present them from a theoretical point of view. Then, we also summarize some applications found in the literature.