Strengthened ordered directionally monotone functions. Links between the different notions of monotonicity

Abstract

In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Lukasiewicz implication.

Directions of directional, ordered directional and strengthened ordered directional increasingness of linear and ordered linear fusion operators

“…There exist some properties that are shared by the three different forms of monotonicity. One of the most relevant ones is that if a function f is increasing along two directions r and s (either directionally, ordered directionally or strengthened ordered directionally), then it increases (in the same sense) along any direction formed by a positive convex combination of r and s. We illustrate this fact in the following three results, which were presented in [7], [5] and [20] Let us now present two additional results that show how, once we have a function that satisfies one of the discussed forms of monotonicity, we can construct new functions that satisfy the same type of monotonicity. For example, the arithmetic mean of m different directionally (resp.…”

“…Proposition 7: Let us now present the ordered linear fusion functions analogue result of Proposition 5. The proof of this fact can be found in [20].…”

“…Recently, two additional forms of monotonicity have been introduced for which the direction of increasingness varies depending on the specific point to aggregate. These forms of monotonicity are called ordered directional (OD) monotonicity and strengthened ordered directional (SOD) monotonicity and were introduced in [5] and [20], respectively. In both forms, the relative size of each input affects the direction along which an ordered directionally, or strengthened ordered directionally, increasing function increases.…”

Local Properties of Strengthened Ordered Directional and Other Forms of Monotonicity

“…Based on the concept of OD monotonicity, in [19] the concept of strengthened ordered directional monotonicity was introduced.…”

“…There also exist other generalizations of monotonicity, such as ordered directional (OD) monotonicity [6] and strengthened ordered directional (SOD) monotonicity [19]. These two concepts are based on directional monotonicity but the direction of increasingness varies from one point of the domain to another, depending on the relative sizes of the components of the specific point.…”