# Abstract:

The present paper is devoted to the transition from crisp domains of probability to fuzzy domains of probability. First, we start with a simple transportation problem and present its solution. The solution has a probabilistic interpretation and it illustrates the transition from classical random variables to fuzzy random variables in the sense of Gudder and Bugajski. Second, we analyse the process of fuzzification of classical crisp domains of probability within the category $ID$ of $D$-posets of fuzzy sets and put into perspective our earlier results concerning categorical aspects of fuzzification. For example, we show that (within $ID$) all nontrivial probability measures have genuine fuzzy quality and we extend the corresponding fuzzification functor to an epireflector. Third, we extend the results to simplex-valued probability domains. In particular, we describe the transition from crisp simplex-valued domains to fuzzy simplex-valued domains via a "simplex'' modification of the fuzzification functor. Both, the fuzzy probability and the simplex-valued fuzzy probability is in a sense minimal extension of the corresponding crisp probability theory which covers some quantum phenomenon.

# Keywords:

fuzzy random variable, domain of probability, crisp random event, fuzzy observable, fuzzification, category of $ID$-poset, epireflection, simplex-valued domains

60A86, 60A05

# References:

1. S.~Bugajski: Statistical maps I. Basic properties. Math. Slovaca 51 (2001), 321-342.   CrossRef
2. S.~Bugajski: Statistical maps II. Basic properties. Math. Slovaca 51 (2001), 343-361.   CrossRef
3. F.~Chovanec and R.~Fri\v c: States as morphisms. Internat. J. Theoret. Phys. 49 (2010), 3050-3100.   CrossRef
4. F.~Chovanec and F.~K\^{o}pka: $D$-posets. In: Handbook of Quantum Logic and Quantum Structures: Quantum Structures. (K.~Engesser, D. M.~Gabbay and D.~Lehmann, eds.), Elsevier, Amsterdam 2007, pp. 367-428.   CrossRef
5. A.~Dvure\v{c}enskij and S.~Pulmannov\'a: New Trends in Quantum Structures. Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava 2000.   CrossRef
6. R.~Fri\v{c}: Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mt. Math. Publ. 30 (2005), 21-34.   CrossRef
7. R.~Fri\v{c}: Statistical maps: a categorical approach. Math. Slovaca 57 (2007), 41-57.   CrossRef
8. R.~Fri\v c: Extension of domains of states. Soft Comput. 13 (2009), 63-70.   CrossRef
9. R.~Fri\v{c}: Simplex-valued probability. Math. Slovaca 60 (2010), 607-614.   CrossRef
10. R.~Fri\v{c}: States on bold algebras: Categorical aspects. J. Logic Comput. (To appear). DOI:10.1093/logcom/exp014   CrossRef
11. R.~Fri\v c and M.~Pap\v co: On probability domains. Internat. J. Theoret. Phys. 49 (2010), 3092-3063.   CrossRef
12. R.~Fri\v c and M.~Pap\v co: A categorical approach to probability theory. Studia Logica 94 (2010), 215-230.   CrossRef
13. S.~Gudder: Fuzzy probability theory. Demonstratio Math. 31 (1998), 235-254.   CrossRef
14. F.~K\^{o}pka and F.~Chovanec: D-posets. Math. Slovaca 44 (1994), 21-34.   CrossRef
15. R.~Mesiar: Fuzzy sets and probability theory. Tatra Mt. Math. Publ. 1 (1992), 105-123.   CrossRef
16. M.~Pap{\v c}o: On measurable spaces and measurable maps. Tatra Mt. Math. Publ. 28 (2004), 125-140.   CrossRef
17. M.~Pap{\v c}o: On fuzzy random variables: examples and generalizations. Tatra Mt. Math. Publ. 30 (2005), 175-185.   CrossRef
18. M.~Pap{\v c}o: On effect algebras. Soft Comput. 12 (2007), 26-35.   CrossRef
19. B.~Rie\v can and D.~Mundici: Probability on $MV$-algebras. In: Handbook of Measure Theory, Vol. II (E.~Pap, ed.), North-Holland, Amsterdam 2002, pp. 869-910.   CrossRef
20. L. A.~Zadeh: Probability measures of fuzzy events. J. Math. Anal. Appl. 23 (1968), 421-427.   CrossRef