Representing preorders with injective monotones

Area of Science:

• Decision Theory
• Mathematical Economics
• Information Theory

Background:

• Real-valued monotones are crucial for understanding preferences in preordered spaces.
• Existing classifications rely on strict monotones and countable multi-utilities.
• Further refinement of these classifications is needed for advanced applications.

Purpose of the Study:

• Introduce and define injective monotones as a new class of real-valued functions.
• Establish the position of preorders admitting injective monotones within existing classifications.
• Extend known results from strict monotones to this new class.

Main Methods:

• Develop a novel class of real-valued functions termed injective monotones.
• Analyze the properties and existence conditions for injective monotones in preordered spaces.
• Construct injective monotones from countable multi-utilities.

Main Results:

• Demonstrate that injective monotones exist for a class of preorders situated between those with strict monotones and countable multi-utilities.
• Generalize key results of strict monotones (Richter-Peleg functions) to injective monotones.
• Establish connections between injective monotones, Debreu denseness, order separability, Shannon entropy, and the uncertainty preorder.

Conclusions:

• Injective monotones offer a refined classification of preordered spaces.
• These monotones provide new insights into the relationship between information theory and decision theory.
• Generalizing Jaynes' maximum entropy principle using injective monotones has implications for statistical inference and machine learning regularization.