Curve-based monotonicity: a generalization of directional monotonicity

Abstract

In this work we propose a generalization of the notion of directional monotonicity. Instead of considering increasingness or decreasingness along rays, we allow more general paths defined by curves in the n-dimensional space. These considerations lead us to the notion of α-monotonicity, where α is the corresponding curve. We study several theoretical properties of α-monotonicity and relate it to other notions of monotonicity, such as weak monotonicity and directional monotonicity.

n-K-Increasing Aggregation Functions

“…, 0)}, is called a pre-aggregation function [16]. There are also known some other concepts of monotonicity, for example, ordered directional monotonicity [17], cone monotonicity [18], or curve monotonicity [19]. However, except for the standard monotonicity of real n-ary functions, none of the mentioned types of monotonicity can be examined by means of the monotonicity in individual variables.…”

Description and Properties of Curve-Based Monotone Functions

“…If a function f is both α-increasing and α-decreasing for a given curve α : [0, 1] → R n , then f is said to be α-constant. For curves defined on an open interval, see [9].…”

“…In this work, we discuss a generalization of directional monotonicity: curvebased monotonicity [9]. Rather than directions given by vectors, curve-based monotonicity studies the monotonicity of functions along general curves α : [0, 1] → R n .…”