FGCounts of Fuzzy Sets with Triangular Norms

Fuzzy Control ◽  
2000 ◽  
pp. 121-131 ◽  
Author(s):  
Maciej Wygralak ◽  
Daniel Pilarski
Keyword(s):  
Fuzzy Sets ◽  
Triangular Norms

scholarly journals On Fuzzy Negations Generated by Fuzzy Implications

2020 ◽  
Vol 28 (1) ◽  
pp. 121-128
Author(s):  
Adam Grabowski
Keyword(s):  
Fuzzy Sets ◽  
Uncertain Information ◽  
Triangular Norms ◽  
Fuzzy Implications ◽  
Formal Framework ◽  
System 2 ◽  
Mizar Mathematical Library ◽  
Fuzzy Connectives

SummaryWe continue in the Mizar system [2] the formalization of fuzzy implications according to the book of Baczyński and Jayaram “Fuzzy Implications” [1]. In this article we define fuzzy negations and show their connections with previously defined fuzzy implications [4] and [5] and triangular norms and conorms [6]. This can be seen as a step towards building a formal framework of fuzzy connectives [10]. We introduce formally Sugeno negation, boundary negations and show how these operators are pointwise ordered. This work is a continuation of the development of fuzzy sets [12], [3] in Mizar [7] started in [11] and partially described in [8]. This submission can be treated also as a part of a formal comparison of fuzzy and rough approaches to incomplete or uncertain information within the Mizar Mathematical Library [9].

CLASSES OF INTUITIONISTIC FUZZY T-NORMS SATISFYING THE RESIDUATION PRINCIPLE

2003 ◽  
Vol 11 (06) ◽  
pp. 691-709 ◽  
Author(s):  
GLAD DESCHRIJVER ◽  
ETIENNE E. KERRE
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Intuitionistic Fuzzy Sets ◽  
Membership Degree ◽  
Important Class ◽  
Triangular Norms ◽  
Intuitionistic Fuzzy

Intuitionistic fuzzy sets constitute an extension of fuzzy sets: while fuzzy sets give a degree to which an element belongs to a set, intuitionistic fuzzy sets give both a membership degree and a non-membership degree. The only constraint on those two degrees is that the sum must be smaller than or equal to 1. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In the fuzzy case a t-norm satisfies the residuation principle if and only if it is left-continuous. Deschrijver, Cornelis and Kerre proved that for intuitionistic fuzzy t-norms the equivalence between the residuation principle and intuitionistic fuzzy left-continuity only holds for t-representable t-norms.1 In this paper we construct particular subclasses of intuitionistic fuzzy t-norms that satisfy the residuation principle but that are not t-representable and we show that a continuous intuitionistic fuzzy t-norm [Formula: see text] satisfying the residuation principle is t-representable if and only if [Formula: see text].

SMETS-MAGREZ AXIOMS FOR R-IMPLICATORS IN INTERVAL-VALUED AND INTUITIONISTIC FUZZY SET THEORY

2005 ◽  
Vol 13 (04) ◽  
pp. 453-464 ◽  
Author(s):  
GLAD DESCHRIJVER ◽  
ETIENNE E. KERRE
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Intuitionistic Fuzzy Set ◽  
Membership Degree ◽  
Triangular Norms ◽  
Intuitionistic Fuzzy ◽  
Special Lattice ◽  
Interval Valued

Interval-valued fuzzy sets constitute an extension of fuzzy sets which give an interval approximating the "real" (but unknown) membership degree. Interval-valued fuzzy sets are equivalent to intuitionistic fuzzy sets in the sense of Atanassov which give both a membership degree and a non-membership degree, whose sum must be smaller than or equal to 1. Both are equivalent to L-fuzzy sets w.r.t. a special lattice L*. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In a previous paper5 we gave a construction for t-norms on L* satisfying the residuation principle which are not t-representable. In this paper we investigate the Smets-Magrez axioms and some other properties for the residual implicator generated by such t-norms.

ORDINAL SUMS IN INTERVAL-VALUED FUZZY SET THEORY

2005 ◽  
Vol 01 (02) ◽  
pp. 243-259 ◽  
Author(s):  
GLAD DESCHRIJVER
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Membership Degree ◽  
Triangular Norms ◽  
Ordinal Sum ◽  
Sum Theorem ◽  
The Universe ◽  
Interval Valued

Interval-valued fuzzy sets form an extension of fuzzy sets which assign to each element of the universe a closed subinterval of the unit interval. This interval approximates the "real", but unknown, membership degree. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. A method for constructing t-norms that satisfy the residuation principle is by using the ordinal sum theorem. In this paper, we construct the ordinal sum of t-norms on [Formula: see text], where [Formula: see text] is the underlying lattice of interval-valued fuzzy set theory, in such a way that if the summands satisfy the residuation principle, then the ordinal sum does too.

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