Weak and directional monotonicity of functions on Riesz spaces to fuse uncertain data

In the theory of aggregation, there is a trend towards the relaxation of the axiom of monotonicity and also towards the extension of the definition to other domains besides real numbers. In this work, we join both approaches by defining the concept of directional monotonicity for functions that take values in Riesz spaces. Additionally, we adapt this notion in order to work in certain convex sublattices of a Riesz space, which makes it possible to define the concept of directional monotonicity for functions whose purpose is to fuse uncertain data coming from type-2 fuzzy sets, fuzzy multisets, n-dimensional fuzzy sets, Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets, among others. Focusing on the latter, we characterize directional monotonicity of interval-valued representable functions in terms of standard directional monotonicity.