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On Probability Domains IV

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Abstract

Stressing a categorical approach, we continue our study of fuzzified domains of probability, in which classical random events are replaced by measurable fuzzy random events. In operational probability theory (S. Bugajski) classical random variables are replaced by statistical maps (generalized distribution maps induced by random variables) and in fuzzy probability theory (S. Gudder) the central role is played by observables (maps between probability domains). We show that to each of the two generalized probability theories there corresponds a suitable category and the two resulting categories are dually equivalent. Statistical maps and observables become morphisms. A statistical map can send a degenerated (pure) state to a non-degenerated one —a quantum phenomenon and, dually, an observable can map a crisp random event to a genuine fuzzy random event —a fuzzy phenomenon. The dual equivalence means that the operational probability theory and the fuzzy probability theory coincide and the resulting generalized probability theory has two dual aspects: quantum and fuzzy. We close with some notes on products and coproducts in the dual categories.

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Acknowledgments

This work was supported by VEGA 2/0031/15.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovak Republic

    Roman Frič

  2. Catholic University in Ružomberok, Hrabovská cesta 1, 034 01, Ružomberok, Slovak Republic

    Martin Papčo

  3. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovak Republic

    Martin Papčo

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  1. Roman Frič
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  2. Martin Papčo
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Correspondence to Martin Papčo.

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Frič, R., Papčo, M. On Probability Domains IV. Int J Theor Phys 56, 4084–4091 (2017). https://doi.org/10.1007/s10773-017-3438-z

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  • DOI: https://doi.org/10.1007/s10773-017-3438-z

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