Sugeno integral in a finite Boolean algebra

The aim of this paper is to provide a representation theorem of the Sugeno integral when the evaluation scale is a finite Boolean algebra. This result is a generalisation of a result proved when the evaluation scale is totally ordered. A major difficulty is the renunciation of the comonotonic functions which have to be replaced by the co-included functions. So we need to care about the properties satisfied by the Sugeno integral when the evaluation scale is a finite totally ordered set. To begin we show that, when the scale is totally ordered, the co-included functions are solely needed to characterise the Sugeno integral and that they are less constraining than the comonotonic functions classically used. Next we focus on finite Boolean algebra, and in this new context we define the Sugeno integral and we present the properties still satisfied. To end this article, we present a representation theorem of the Sugeno integral on a finite Boolean algebra.