Editing Fuzzy set (section)

===''L''-fuzzy sets===
Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) [[algebraic structure|algebra]] or [[structure (mathematical logic)|structure]] <math>L</math> of a given kind; usually it is required that <math>L</math> be at least a [[poset]] or [[lattice (order)|lattice]]. These are usually called '''''L''-fuzzy sets''', to distinguish them from those valued over the unit interval. The usual membership functions with values in [0,&thinsp;1] are then called [0,&thinsp;1]-valued membership functions. These kinds of generalizations were first considered in 1967 by [[Joseph Goguen]], who was a student of Zadeh.<ref>{{cite journal | doi=10.1016/0022-247X(67)90189-8 | title=L-fuzzy sets | date=1967 | last1=Goguen | first1=J.A |author-link=Joseph Goguen | journal=Journal of Mathematical Analysis and Applications | volume=18 | pages=145–174 }}</ref> A classical corollary may be indicating truth and membership values by {f,&thinsp;t} instead of {0,&thinsp;1}.

An extension of fuzzy sets has been provided by [[Krassimir Atanassov|Atanassov]]. An '''intuitionistic fuzzy set''' (IFS) <math>A</math> is characterized by two functions:
:1. <math>\mu_A(x)</math> – degree of membership of ''x''
:2. <math>\nu_A(x)</math> – degree of non-membership of ''x''
with functions <math>\mu_A, \nu_A: U \to [0,1]</math> with <math>\forall x \in U: \mu_A(x) + \nu_A(x) \le 1</math>.

This resembles a situation like some person denoted by <math>x</math> voting 
* for a proposal <math>A</math>: (<math>\mu_A(x)=1, \nu_A(x)=0</math>), 
* against it: (<math>\mu_A(x)=0, \nu_A(x)=1</math>), 
* or abstain from voting: (<math>\mu_A(x)=\nu_A(x)=0</math>).
After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions.

For this situation, special "intuitive fuzzy" negators, t- and s-norms can be defined. With <math>D^* = \{(\alpha,\beta) \in [0, 1]^2 : \alpha + \beta = 1 \}</math> and by combining both functions to <math>(\mu_A,\nu_A): U \to D^*</math> this situation resembles a special kind of ''L''-fuzzy sets.

Once more, this has been expanded by defining '''picture fuzzy sets''' (PFS) as follows: A PFS A is characterized by three functions mapping ''U'' to [0,&thinsp;1]: <math>\mu_A, \eta_A, \nu_A</math>, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition <math>\forall x \in U: \mu_A(x) + \eta_A(x) + \nu_A(x) \le 1</math>
This expands the voting sample above by an additional possibility of "refusal of voting".

With <math>D^* = \{(\alpha,\beta,\gamma) \in [0, 1]^3 : \alpha + \beta + \gamma = 1 \}</math> and special "picture fuzzy" negators, t- and s-norms this resembles just another type of ''L''-fuzzy sets.<ref>Bui Cong Cuong, Vladik Kreinovich, Roan Thi Ngan: [http://digitalcommons.utep.edu/cgi/viewcontent.cgi?article=2050&context=cs_techrep A classification of representable t-norm operators for picture fuzzy sets], in: Departmental Technical Reports (CS). Paper 1047, 2016</ref>