Fuzzification of crisp domains

Frič, Roman, and Papčo, Martin. "Fuzzification of crisp domains." Kybernetika 46.6 (2010): 1009-1024. <http://eudml.org/doc/196793>.

@article{Frič2010,
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.},
author = {Frič, Roman, Papčo, Martin},
journal = {Kybernetika},
keywords = {domain of probability; fuzzy random variable; crisp random event; fuzzy observable; fuzzification; category of $ID$-poset; epireflection; simplex-valued domains; domain of probability; fuzzy random variable; crisp random event; fuzzy observable; fuzzification; category of -poset; simplex-valued domains},
language = {eng},
number = {6},
pages = {1009-1024},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Fuzzification of crisp domains},
url = {http://eudml.org/doc/196793},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Frič, Roman
AU - Papčo, Martin
TI - Fuzzification of crisp domains
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 6
SP - 1009
EP - 1024
AB - 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.
LA - eng
KW - domain of probability; fuzzy random variable; crisp random event; fuzzy observable; fuzzification; category of $ID$-poset; epireflection; simplex-valued domains; domain of probability; fuzzy random variable; crisp random event; fuzzy observable; fuzzification; category of -poset; simplex-valued domains
UR - http://eudml.org/doc/196793
ER -