Nilpotent T-norms Research Articles

Difference operators on fuzzy sets

Based on the properties of the difference operator on crisp sets, a fuzzy difference operator in fuzzy logic is defined as a continuous binary operator on the closed unit interval with some boundary conditions. In this paper, the structures and properties of fuzzy difference operators are studied. The main results are: (1) Using the axiomatic approach, some generalizations of classical tautologies for fuzzy difference operators are obtained. (2) Based on the model theoretic approach, the fuzzy difference operator constructed by a nilpotent t-norm and a strong negation is characterized. (3) the paper discusses the relationship between the fuzzy difference operator and symmetric difference operator which was raised in [3].

The weak dominance between t-norms and t-conorms revisited

The weak dominance relation between two binary operations was introduced as an extension of the dominance relation and the modularity equation. This paper continues the study of the weak dominance between t-norms and t-conorms. First, we present the characterization of the weak dominance of a nilpotent t-norm over a continuous Archimedean t-norm. Second, we establish the generalized sufficient and necessary conditions for the weak dominance of continuous t-norms over continuous t-norms. Finally, we obtain a generalized characterization of a continuous t-conorm weakly dominating a continuous t-norm by showing that SM weakly dominates any t-norm.

An analysis of the invariance property with respect to powers of nilpotent t-norms on fuzzy implication functions

The invariance property with respect to powers of continuous t-norms is an additional property of fuzzy implication functions which is particularly interesting in the area of approximate reasoning. In this paper, we provide the characterization of all the binary functions which are invariant with respect to the powers of a nilpotent t-norm and, from the restriction of this result to the definition of fuzzy implication function, we introduce the family of nilpotent T-power invariant implications. We study when the members of this family do satisfy other additional properties apart from the invariance. Moreover, the intersection between the family of nilpotent T-power invariant implications and several families of fuzzy implication functions is investigated.

On some classes of nullnorms and h-pseudo homogeneity

Nullnorms are aggregation functions which generalize t-norms and t-conorms. For each nullnorm there exists an annihilator element such that, below it, the function behaves like a t-conorm, and, above it, like a t-norm. In this paper, we study some classes of nullnorms which naturally arise from well known classes of t-norms and t-conorms, such as idempotent, Archimedean, cancellative, positive and nilpotent t-norms and t-conorms. We also present the concept of h-pseudo-homogeneous nullnorm and we study when these classes of nullnorms fullfill it.

Characterizations of Fuzzy Implications Generated by Continuous Multiplicative Generators of T-Norms

In this article, we define the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications by continuous multiplicative generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms, along with the research lines of Yager's <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$f$</tex-math></inline-formula> -generated implications by continuous additive generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$f$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g$</tex-math></inline-formula> -generated implications by continuous additive generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -conorms and of Balasubramaniam's <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> -generated implications by continuous multiplicative generators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -conorms. This article is mainly motivated by many desirable properties of such generator generating implications for their flexible ordering property, unique determinations by induced natural negations, close relations to nilpotent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> -conditionality, the law of importation with more choices of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> other than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T_{P}$</tex-math></inline-formula> , and good distributivity over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -norms and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t$</tex-math></inline-formula> -conorms. The relationships of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications to other well-known classes of fuzzy implications are clarified: they will unify <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g$</tex-math></inline-formula> -generated implications with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g(1)&lt; \infty$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> -generated implications with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h(1)&gt;0$</tex-math></inline-formula> and hence, have intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(S,N)$</tex-math></inline-formula> -implications; the only intersection with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R$</tex-math></inline-formula> -implications is the Goguen implication and they have no intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$f$</tex-math></inline-formula> -generated implications. The main results of this article are several characterizing theorems for (sometimes subclasses of) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications by means of aforementioned desirable properties from the respective perspectives, of which some partially solved or enriched related open problems in the literature. Finally, the superiorities of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -generated implications for modeling strict fuzzy preference relations and for constructing novel fuzzy reasoning methods are analyzed.

Formula omitted]-implications and distributivity of implications on [formula omitted] over t-representable t-norms: The case of strict and nilpotent t-norms

In this paper, a new class of intuitionistic fuzzy implications (IFIs) known as (fI,ω)-implications is introduced which is a generalized form of Yager’s f-implications in intuitionistic fuzzy environment (IFE). The basic properties of these implications are discussed in detail. It is shown that (fI,ω)-implications are not only the generalizations of Yager’s f-implications, but also the generalizations of R-, (S,N)- and QL-implications in IFE. The distributivity equations II(T(u,v),w)=S(II(u,w),II(v,w)) and II(u,T1(v,w))=T2(II(u,v),II(u,w)) over t-representable t-norms and t-conorms generated from nilpotent and strict t-norms in IF set theory are discussed. Also, we solve the open problems concerning characterize all of the correct solutions of the distributive equation II(u,T1(v,w))=T2(II(u,v),II(u,w)) when t-norms are strict and nilpotent.

On the isomorphy of categories of probabilistic limit spaces under t-norms

We show that for the classes of strict t-norms, the categories of probabilistic limit spaces under these t-norms are all isomorphic to each other. The same is true for the categories of probabilistic limit spaces under nilpotent t-norms. To show this, we study the isomorphisms between the categories of probabilistic limit spaces under a t-norm, limit tower spaces and approach limit spaces. Similar results are obtained for probabilistic Cauchy spaces and probabilistic uniform limit spaces.

Distributivity of implication operations over t-representable t-norms in interval-valued fuzzy set theory: The case of nilpotent t-norms

In this article we discuss the distributive equation of implications I(x,T1(y,z))=T2(I(x,y),I(x,z)) over t-representable t-norms generated from nilpotent t-norms in interval-valued fuzzy sets theory. As a byproduct result we show all solutions of some functional equation related to this case.

Morphological analysis of 3D SPECT images via nilpotent t-norms in diagnosis of Alzheimer's disease

We propose a fuzzy image processing approach serving as a potential diagnostic tool for Alzheimer's disease. We have set up a sequence of several image-processing methods based on morphological fuzzy edge detection followed by a region-growing segmentation. We applied these operations to two sets of three-dimensional SPECT images of human brains: one set of images was composed of patients with Alzheimer's disease and the second was represented by control brains from non-demented patients. Then we undertook an analysis of nilpotent t-norms forming the edge detectors and the parameter of Gaussian filter, and in the end we carried out an evaluation of segments that was computed after the final watershed segmentation. This article provides a detailed description of these methods as well as the analysis of evaluated data using Student's two-sample t-test. Traditional gradient magnitude edge detectors are included and compared with fuzzy detectors. Our goal was to demonstrate the usefulness of the Łukasiewicz BL-algebra in feature extraction of 3D biomedical images by using enhanced methods of image morphology.

Differences between t-norms in fuzzy control

The application of conjunctive aggregation functions in fuzzy control systems with n inputs is discussed, and the effect of the choice of a continuous t-norm in the inference phase for Takagi–Sugeno–Kang (TSK) systems is computed. A continuous t-norm modeling AND connective in antecedent part of fuzzy rules can be reduced just to strict or nilpotent t-norm. The isomorphism of strict (nilpotent) t-norms enables simpler fitting of TSK fuzzy system parameters and reduces the computational complexity. Similar principle can also be used in the case of some noncommutative conjunctive aggregation functions modeling AND connective. The effect of the choice of a continuous t-norm is then evaluated on well-known case studies in fuzzy control, the Sinc function, and the urban traffic noise control system.

Real vector space of LR-fuzzy intervals with respect to the shape-preserving t-norm-based addition

The construction of the membership function of fuzzy intervals is an important problem in vagueness modeling. The preservation of the shape of fuzzy sets during the addition is a natural requirement in practical computation. The LR-fuzzy intervals introduced by Dubois and Prade, satisfy this requirement if the addition is based on the nilpotent t-norm, generated by L or R shape functions. The shortcoming that not any LR-fuzzy interval has an opposite (inverse related to shape-preserving t-norm-based addition), can be solved, if the set of LR-fuzzy intervals is isomorphically included in an extended set, and this extended set forms a group with respect to shape-preserving t-norm-based addition. In this paper we construct the extended set of these LR-fuzzy intervals. We also show that the extended set is a real vector space with scalar product, and the modal intervals can be considered as the elements of this extended set. Finally, we present the algebraic form of LR-fuzzy intervals and the associated application.

Continuous R-implications generated from representable aggregation functions

This paper deals with a new kind of residual fuzzy implications derived from non-associative aggregation functions. Some properties of such implications, including their expression through a continuous increasing unary function, are studied in detail. This allows for new families of continuous fuzzy implications satisfying the modus ponens for a given nilpotent t-norm.

Convex combinations of nilpotent triangular norms

In this paper we deal with the open problem of convex combinations of continuous triangular norms stated by Alsina, Frank, and Schweizer [C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128–140, Problems 5 and 6]. They pose a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. The main result of this paper gives a negative answer to the question for any pair of continuous Archimedean triangular norms with different supports. With the help of this result we show that a non-trivial convex combination of nilpotent t-norms is never a t-norm. The main result also gives an alternative proof to the result presented by Ouyang and Fang [Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., 68 (11) (2008) 3382–3387, Theorem 3.1]. In proof of the main theorem we utilize the Reidmeister condition known from the web geometry.

Triangular norms. Position paper I: basic analytical and algebraic properties

We present the basic analytical and algebraic properties of triangular norms. We discuss continuity as well as the important classes of Archimedean, strict and nilpotent t-norms. Triangular conorms and De Morgan triples are also mentioned. Finally, a brief historical survey on triangular norms is given.

Residuated fuzzy logics with an involutive negation

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, for t-norms without non-trivial zero divisors, $\neg$ is Godel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation.

A note on negations and nilpotent t-norms

In this paper, we explore general relationships among negations, convex Archimedean nilpotent t-norms, and automorphisms of the unit interval I . Each nilpotent t-norm has a (strong) negation naturally associated with it, namely, η △(x)=⋁{y∈[0,1]:x△y=0} . The same negation is determined by the formula η △(x)=f −1(f(0)/f(x)) where f is a (multiplicative) generating function for the t-norm △. A system ( I,△,▿,∗) is called de Morgan if x▿y=(x ∗△y ∗) ∗ ; Stone if x△y=0 if and only if y⩽x ∗ , and x ∗▿x ∗∗=1 ; and Boolean if it is both de Morgan and Stone. A system is shown to be Boolean if and only if ∗=η △ and x▿y=η △(η △ x)△η △(y)) . We also look at de Morgan, weak Boolean and Stone systems on the lattice I [2]={(x,y)∈ I× I:x⩽y} and compare properties of related systems on I and on I [2] .