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Representing preorders with injective monotones

Author

Listed:
  • Pedro Hack

    (Institute of Neural Information Processing, Ulm University)

  • Daniel A. Braun

    (Institute of Neural Information Processing, Ulm University)

  • Sebastian Gottwald

    (Institute of Neural Information Processing, Ulm University)

Abstract

We introduce a new class of real-valued monotones in preordered spaces, injective monotones. We show that the class of preorders for which they exist lies in between the class of preorders with strict monotones and preorders with countable multi-utilities, improving upon the known classification of preordered spaces through real-valued monotones. We extend several well-known results for strict monotones (Richter–Peleg functions) to injective monotones, we provide a construction of injective monotones from countable multi-utilities, and relate injective monotones to classic results concerning Debreu denseness and order separability. Along the way, we connect our results to Shannon entropy and the uncertainty preorder, obtaining new insights into how they are related. In particular, we show how injective monotones can be used to generalize some appealing properties of Jaynes’ maximum entropy principle, which is considered a basis for statistical inference and serves as a justification for many regularization techniques that appear throughout machine learning and decision theory.

Suggested Citation

  • Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2022. "Representing preorders with injective monotones," Theory and Decision, Springer, vol. 93(4), pages 663-690, November.
    • Handle: RePEc:kap:theord:v:93:y:2022:i:4:d:10.1007_s11238-021-09861-w
    DOI: 10.1007/s11238-021-09861-w
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    Cited by:

    1. Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2023. "The classification of preordered spaces in terms of monotones: complexity and optimization," Theory and Decision, Springer, vol. 94(4), pages 693-720, May.

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