The classification of preordered spaces in terms of monotones: 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.

References in corpus (8)

• Conditions for a class of entanglement transformations
• The second laws of quantum thermodynamics
• The Physics and Mathematics of the Second Law of Thermodynamics
• Necessary and Sufficient Conditions for the Trumping Relation
• Catalytic Conversion Probabilities for Bipartite Pure States
• Time functions as utilities
• A topological study for the existence of lower-semicontinuous Richter-Peleg multi-utilities
• The infinite information gap between mathematical and physical representations

Cited by in corpus (2)

• Majorization requires infinitely many second laws
• Countability constraints in order-theoretic approaches to computability