The classification of preordered spaces in terms of monotones: complexity and optimization

TL;DR

This paper classifies preordered spaces based on the existence and number of real-valued monotones, providing new insights into the complexity and optimization of decision spaces.

Contribution

It introduces a comprehensive classification of preordered spaces using recent monotone classes, enhancing understanding of their complexity and optimization.

Findings

New classification of preordered spaces based on monotone cardinality

Characterization of monotones via separating families of increasing sets

Clarification of the relationship between complexity and optimization

Abstract

The study of complexity and optimization in decision theory involves both partial and complete characterizations of preferences over decision spaces in terms of real-valued monotones. With this motivation, and following the recent introduction of new classes of monotones, like injective monotones or strict monotone multi-utilities, we present the classification of preordered spaces in terms of both the existence and cardinality of real-valued monotones and the cardinality of the quotient space. In particular, we take advantage of a characterization of real-valued monotones in terms of separating families of increasing sets in order to obtain a more complete classification consisting of classes that are strictly different from each other. As a result, we gain new insight into both complexity and optimization, and clarify their interplay in preordered spaces.