Extremal Completions of Triangular Norms Known on a Subregion of the Unit Interval

2015 ◽  
pp. 21-32 ◽  
Author(s):  
Andrea Mesiarová-Zemánková
Keyword(s):  
Unit Interval ◽  
Triangular Norms

REVISITING Xor-IMPLICATIONS: CLASSES OF FUZZY (CO)IMPLICATIONS BASED ON f-Xor (f-XNor) CONNECTIVES

2013 ◽  
Vol 21 (06) ◽  
pp. 899-925 ◽  
Author(s):  
BENJAMÍN BEDREGAL ◽  
RENATA HAX SANDER REISER ◽  
GRAÇALIZ PEREIRA DIMURO
Keyword(s):  
Boundary Conditions ◽  
Unit Interval ◽  
End Points ◽  
Triangular Norms ◽  
Fuzzy Implications ◽  
Classical Definition ◽  
Definition Of ◽  
Exclusive Or ◽  
Intrinsic Definition ◽  
Fuzzy Connectives

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.

ON A CLASS OF UNINORMS

2002 ◽  
Vol 10 (supp01) ◽  
pp. 5-10 ◽  
Author(s):  
JÓZEF DREWNIAK ◽  
PAWEŁ DRYGAŚ
Keyword(s):  
Unit Interval ◽  
Triangular Norms ◽  
Binary Operations

Paper deals with binary operations in unit interval. We investigate connections between families of triangular norms, triangular conorms and uninorms. Certain structures of uninorms admits only the idempotent case.

The Relations between Implications and Left (Right) Semi-Uninorms on a Complete Lattice

2015 ◽  
Vol 23 (02) ◽  
pp. 245-261 ◽  
Author(s):  
Xiaoying Hao ◽  
Meixia Niu ◽  
Zhudeng Wang
Keyword(s):  
Complete Lattice ◽  
Unit Interval ◽  
Neutral Element ◽  
Order Property ◽  
Triangular Norms

Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we study the relations between implications and left (right) semi-uninorms on a complete lattice. We firstly investigate the left (right) semi-uninorms induced by implications, give some conditions such that the operations induced by implications constitute left or right semi-uninorms, and demonstrate that the operations induced by a right infinitely ∧-distributive implication, which satisfies the order property, are left (right) infinitely ∨-distributive left (right) semi-uninorms. Then, we discuss the residual operations of left (right) semi-uninorms and show that left (right) residual operators of strict left (right)-conjunctive left (right) infinitely ∨-distributive left (right) semi-uninorms are right infinitely ∧-distributive implications that satisfy the order property. Finally, we reveal the relationships between strict left (right)-conjunctive left (right) infinitely ∨-distributive left (right) semi-uninorms and right infinitely ∧-distributive implications which satisfy the order property.

scholarly journals Quasi-homogeneous associative functions

1998 ◽  
Vol 21 (2) ◽  
pp. 351-357 ◽  
Author(s):  
Bruce R. Ebanks
Keyword(s):  
Special Kind ◽  
Unit Interval ◽  
Space Theory ◽  
Triangular Norm ◽  
Probabilistic Metric Space ◽  
Triangular Norms ◽  
Homogeneity Property ◽  
Associative Function ◽  
Probabilistic Metric ◽  
Associative Functions

A triangular norm is a special kind of associative function on the closed unit interval[0,1]. Triangular norms (ort-norms) were introduced in the context of probabilistic metric space theory, and they have found applications also in other areas, such as fuzzy set theory. We determine the explicit forms of allt-norms which satisfy a generalized homogeneity property called quasi-homogeneity.

scholarly journals Archimedean components of triangular norms

2005 ◽  
Vol 78 (2) ◽  
pp. 239-255 ◽  
Author(s):  
Erich Peter Klement ◽  
Radko Mesiar ◽  
Endre Pap
Keyword(s):  
Unit Interval ◽  
Neutral Element ◽  
Ordered Semigroup ◽  
Triangular Norms ◽  
Construction Methods

AbstractThe Archimedean components of triangular norms (which turn the closed unit interval into anabelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordinal sums and additive generators, new types of left-continuous triangular norms are constructed.

Continuous completions of triangular norms known on a subregion of the unit interval

2017 ◽  
Vol 308 ◽  
pp. 27-41
Author(s):  
Andrea Mesiarová-Zemánková
Keyword(s):  
Unit Interval ◽  
Triangular Norms

Two-Dimensional Copulas as Important Binary Aggregation Operators

2006 ◽  
Vol 10 (4) ◽  
pp. 522-526
Author(s):  
Endre Pap ◽  
◽  
Marta Takács ◽  
Keyword(s):  
Aggregation Operators ◽  
Unit Interval ◽  
Two Dimensional ◽  
Triangular Norms ◽  
Related Research ◽  
Research Results

We introduce 2-copulas (copulas, shortly) and recent related research results. We present invariant copulas and their application in the theory of aggregation operators. Copulas are transformed by increasing bijections at the unit interval and discuss copula attractors. We also present results on the approximation of associative copulas by strict and nilpotent triangular norms.

On triangular norms representable as ordinal sums based on interior operators on a bounded meet semilattice

2021 ◽  
Author(s):  
Yao Ouyang ◽  
Hua-Peng Zhang ◽  
Zhudeng Wang ◽  
Bernard De Baets
Keyword(s):  
Triangular Norms

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

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