Editing Fuzzy set (section)

==Fuzzy categories<!--'Goguen category' rediretcs here-->==
The use of [[Set theory|set membership]] as a key component of [[category theory]] can be generalized to fuzzy sets. This approach, which began in 1968 shortly after the introduction of fuzzy set theory,<ref>J. A. Goguen "Categories of fuzzy sets: applications of non-Cantorian set theory" PhD Thesis University of California, Berkeley, 1968</ref> led to the development of '''Goguen categories'''<!--boldface per WP:R#PLA--> in the 21st century.<ref>Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations" 2007 [[Springer Verlag|Springer]] {{ISBN|9781402061639}}</ref><ref name=goguencateg>{{cite journal | doi=10.1016/S0165-0114(02)00508-0 | title=Representation theory of Goguen categories | date=2003 | last1=Winter | first1=Michael | journal=Fuzzy Sets and Systems | volume=138 | pages=85–126 }}</ref> In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in ''L''-fuzzy sets.<ref name=goguencateg/><ref>{{cite journal | doi=10.1016/0022-247X(67)90189-8 | title=L-fuzzy sets | date=1967 | last1=Goguen | first1=J.A | journal=Journal of Mathematical Analysis and Applications | volume=18 | pages=145–174 }}</ref>

There are numerous mathematical extensions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965 by Zadeh, many new mathematical constructions and theories treating imprecision, inaccuracy, vagueness, uncertainty and vulnerability have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others attempt to mathematically model inaccuracy/vagueness and uncertainty in a different way. The diversity of such constructions and corresponding theories includes:

* Fuzzy Sets (Zadeh, 1965)
* interval sets (Moore,     1966),
* L-fuzzy sets (Goguen,     1967),
* flou sets (Gentilhomme,     1968),
* type-2 fuzzy sets and     type-n fuzzy sets (Zadeh, 1975),
* interval-valued fuzzy     sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),
* level fuzzy sets     (Radecki, 1977)
* rough sets (Pawlak,     1982),
* intuitionistic fuzzy     sets (Atanassov, 1983),
* fuzzy multisets (Yager,     1986),
* intuitionistic L-fuzzy     sets (Atanassov, 1986),
* rough multisets     (Grzymala-Busse, 1987),
* fuzzy rough sets     (Nakamura, 1988),
* real-valued fuzzy sets     (Blizard, 1989),
* vague sets (Wen-Lung Gau     and Buehrer, 1993),
* α-level sets (Yao,     1997), 
* shadowed sets (Pedrycz,     1998),
* neutrosophic sets (NSs)     (Smarandache, 1998),
* bipolar fuzzy sets     (Wen-Ran Zhang, 1998),
* genuine sets (Demirci,     1999),
* soft sets (Molodtsov,     1999),
* complex fuzzy set     (2002),
* intuitionistic fuzzy     rough sets (Cornelis, De Cock and Kerre, 2003)
* L-fuzzy rough sets     (Radzikowska and Kerre, 2004),
* multi-fuzzy sets (Sabu     Sebastian, 2009),
* generalized rough fuzzy     sets (Feng, 2010)
* rough intuitionistic     fuzzy sets (Thomas and Nair, 2011),
* soft rough fuzzy sets     (Meng, Zhang and Qin, 2011)
* soft fuzzy rough sets     (Meng, Zhang and Qin, 2011)
* soft multisets     (Alkhazaleh, Salleh and Hassan, 2011)
* fuzzy soft multisets     (Alkhazaleh and Salleh, 2012)
* pythagorean fuzzy set     (Yager , 2013), 
* picture fuzzy set     (Cuong, 2013),
* spherical fuzzy set     (Mahmood, 2018).