Non - numeric labels and constrained focal elements

The theory of belief functions allows to build a large family of combination operators, based mostly on intersections and unions between the focal elements expressed by the experts, and multiplications and additions on the masses affected to these focal elements. This chapter explores some algebraic structures where these operators behave differently, masses being linguistic labels, or focal elements being more, or less, than an union of singletons of a discernment space. In some cases, it will be necessary to forget this space and the notion of singleton to work within a space of possible focal elements. We propose five new definitions of labels, with the corresponding algebras, to replace the masses of [0, 1]. Adaptations of the theory of belief functions to six constrained spaces for the focal elements expressed by the experts are presented.