Pointwise aggregation of maps: Its structural functional equation and some applications to social choice theory

Abstract

We study a structural functional equation that is directly related to the pointwise aggregation of a finite number of maps from a nonempty given set into another. First we establish links between pointwise aggregation and invariance properties. Then, paying attention to the particular case of aggregation operators of a finite number of real-valued functions, we characterize several special kinds of aggregation operators, as strictly monotone modifications of projections. As a case study, we introduce a first approach of type-2 fuzzy sets via fusion operators. We develop some applications and possible uses related to the analysis of properties of social evaluation functionals in social choice, showing that those functionals can actually be described by using methods that derive from this setting.

Hyperspectral imaging using notions from type-2 fuzzy sets

“…In [9] it is shown that the pointwise interpretation of functions, which is the most usual one in literature [21], is not appropriate for introducing operations in T2FSs. The significance of the function, as mathematical object, cannot be observed individually at each point of its domain x; it can only be observed, hence, in its indivisible behaviour over the domain.…”

Social evaluation functionals with an arbitrary set of alternatives

“…B defined by FðU ÞðxÞ ¼ max fU ðxÞ; U ðyÞg, and FðU ÞðyÞ ¼ min fU ðxÞ; U ðyÞg, 8 U 2 B. Anyway, the cases #X ¼ 1 or n ¼ 1 lack of interest in the social choice setting [for more details, see DeMiguel et al (2017)].…”