Continuous T-norms Research Articles

Invariant idempotent ⁎-measures generated by iterated function systems

It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces.We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented.

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.

Generating methods of some classes of fuzzy implications obtained by unary functions and algebraic structures

The existing generating methods of fuzzy implications are closed in the set of all fuzzy implications, but not when applied to some families of these operators. In this paper, some binary operations are defined on some well-established families of fuzzy implications. Namely, the families of (S,N)-implications with continuous t-conorms, R-implications obtained from continuous t-norms, Yager's f- and g-generated implications, h- and generalized (h,e)-implications and k-implications are considered and it is proved that with these operations, they are lattices. Moreover, some other lattice substructures are defined in some of the families when some restrictions on the underlying generators of the implications are imposed. Furthermore, it is emphasized that these generating methods preserve important properties like the exchange principle and the law of importation.

On Equivalence Operators Derived from Overlap and Grouping Functions

This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. In this paper, two models of fuzzy equivalence operators are obtained by the composition of overlap functions, grouping functions and fuzzy negations. Their main properties are also studied.

Fuzzy difference operators derived from overlap functions

This paper introduces the concept of (O, N)-difference, for an overlap function O and a fuzzy negation N. (O, N)-differences are weaker than fuzzy difference constructed from positive and continuous t-norms and fuzzy negations, in the sense that (O, N)-differences do not necessarily satisfy certain properties, as the right neutrality principle, but only weaker versions of these properties. This paper analyzes the main properties satisfied by (O, N)-differences, and provides a characterization of (O, N)-difference.

Some notes on possibilistic randomisation with t-norm based joint distributions in strategic-form games

This article continues the investigation started in [18] on the role of possibilistic mixed strategies in strategic-form games. In this earlier work we assumed, as standard in possibility theory, that joint possibility distributions were computed by combining possibilistic mixed strategies with the minimum t-norm. In this paper, we investigate the consequences of defining joint possibility distributions by using any continuous t-norm, with players' expected utilities based on the Choquet integral. We characterise under which conditions a pair of possibilistic mixed strategies is an equilibrium, generalising the results first presented in [18], and also show that the set of equilibria in possibilistic mixed strategies depends on the set of idempotent elements of a t-norm and not just on the chosen t-norm.

Symmetric Difference Operators Derived from Overlap and Grouping Functions

This paper introduces the concept of symmetric difference operators in terms of overlap and grouping functions, for which the associativity property is not strongly required. These symmetric difference operators are weaker than symmetric difference operators in terms of positive and continuous t-norms and t-conorms. Therefore, in the sense of the characters of mathematics, these operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. We analyze several related important properties based on two models of symmetric differences.

Formal balls of Q-categories

The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized to Q-categories by Kostanek and Waszkiewicz, where Q is a commutative and unital quantale. This paper concerns the influence of the structure of the quantale Q on the connection between Yoneda completeness of Q-categories and directed completeness of their sets of formal balls. In the case that Q is the unit interval [0, 1] equipped with a continuous t-norm &, it is shown that in order that Yoneda completeness of each Q-category be equivalent to directed completeness of its set of formal balls, a necessary and sufficient condition is that the t-norm & is Archimedean.

Distributivity Conditions of Idempotent Uninorms and Two Special Kinds of Aggregation Functions

Recently, some authors studied the distributive equations for continuous t-norms and some families of usual classes of uninorms over overlap functions in Refs. 25 and 32, but lacked a complete characterization of the distributivity on idempotent uninorms and overlap or grouping functions widely used in image processing. As a supplement to the previous results, in this article we fully characterize the distributivity equations of idempotent uninorms over these two functions by virtue of the associated functions of idempotent uninorms. Moreover, we also discuss the distributivity conditions of above two special functions over idempotent uninorms, yet find in this case that the associated functions of idempotent uninorms must satisfy particular conditions and those two functions usually are a class of special aggregation functions with a constant domain whose value equals to the neutral element of the idempotent uninorm.

On alpha-cross-migrativity of t-conorms over fuzzy implications

As a weaker form of the classical commuting equation, α-cross-migrativity properties between conjunctive connectives including t-norms, uninorms, and overlap functions have been extensively investigated. However, there has been no comparative analysis of the α-cross-migrativity between disjunctive connectives and fuzzy implications. The work is dedicated to the study of α-cross-migrativity involving t-conorms and fuzzy implications. The investigation is presented in two separate parts: the first part focuses on the case that fuzzy implication satisfies some property, especially, the order property. The second one deals with the situation where fuzzy implication belongs to some special classes, i.e., (S,N)-implications, R-implications, and Yager's implications (i.e., f- and g-generated implications). Some interesting results are obtained: full characterizations of α-cross-migrativity for continuous t-conorms over fuzzy implications satisfying the order property, (S,N)-implications, R-implications induced by border continuous t-norms and Yager's implications. As a by-product, some constructions of ordinal sum t-conorms and ordinal sum fuzzy implications satisfying α-cross-migrativity are obtained. Moreover, some supporting examples for solutions are given.

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.

Generation of continuous T-norms through latticial operations

It is well known that the usual point-wise ordering over the set T of t-norms makes it a poset but not a lattice, i.e., the point-wise maximum or minimum of two t-norms need not always be a t-norm again. In this work, we propose, two binary operations ▪ on the set TCA of continuous Archimedean t-norms and obtain, via these binary operations, a partial order relation ⊑, different from the usual point-wise order ≤, on the set TCA. As an interesting outcome of this structure, some stronger versions of some existing results dealing with the upper and lower bounds of two continuous Archimedean t-norms with respect to the point-wise order ≤ are also obtained. Finally, with the help of the operations ▪ on the set TCA, two binary operations ⊕,⊗ on the set TC of continuous t-norms are proposed and showed that (TC,⊕,⊗) is a lattice. Thus we have both a way of generating continuous t-norms from continuous t-norms and also obtain an order on them.

Notes on “On (IO,O)-fuzzy rough sets based on overlap functions”

Qiao investigated the properties and topological structures of (IO,O)-fuzzy rough sets, which extended the classical conjunction operator in rough approximation operator to an overlap function O. However, there are some faults in the characterizations of (IO,O)-fuzzy rough sets, such as wrong conclusions and strict condition, even if the overlap function O is assumed to be a continuous t-norm with no non-trivial zero divisors. This paper further discusses (IO,O)-fuzzy rough sets and rectifies those faults. Moreover, some examples are presented to show those faults.

The Product Fuzzy Metric Space and its Basic Properties

A new type of FMS, known as a product FMS (or simply p-FMS) is introduced, the reason behind this name is the continuous t-norm used here is binary operation usually product between number in [0, 1]. To show that this type of spaces is exists we give some examples, then some important basic properties, of this space are introduced with its proves. Moreover we study fuzzy continuous and uniformly fuzzy continuous function by proving basic properties of these notions. Furthermore the prove of cross product of finite number of p-FMS is again a p-FMS is introduced.

A New Extension to the Intuitionistic Fuzzy Metric-like Spaces

In this manuscript, we introduce the concept of intuitionistic fuzzy controlled metric-like spaces via continuous t-norms and continuous t-conorms. This new metric space is an extension to intuitionistic fuzzy controlled metric-like spaces, controlled metric-like spaces and controlled fuzzy metric spaces, and intuitionistic fuzzy metric spaces. We prove some fixed-point theorems and we present non-trivial examples to illustrate our results. We used different techniques based on the properties of the considered spaces notably the symmetry of the metric. Moreover, we present an application to non-linear fractional differential equations.

Distributivity characterization of idempotent uni-nullnorms and overlap or grouping functions

Recently, some authors studied the distributive laws of continuous t-norms and some families of the common classes of uninorms over overlap and grouping functions in [33], [41], but until now a complete characterization of the distributivity on idempotent uninorms over overlap or grouping functions widely used in image processing is still unresolved. Moreover, authors in [55] characterized the distributivity equations of uni-nullnorms with continuous Archimedean underlying operators over the above two functions. In this paper, we proceed with the distributivity characterization of idempotent uni-nullnorms over them which obviously generalizes the corresponding results of idempotent uninorms over these two functions. During the process, we introduce a class of weak overlap and grouping functions with weak coefficients, and obtain the full characterizations of the above functions by considering the different values of the underlying uninorms' associated functions of idempotent uni-nullnorms on the interval endpoints and comparing the values with the weak coefficients. These obtained results totally answer the question on the distributive solutions of idempotent uninorms over overlap functions, which have been mentioned as their future works in [33]. In additions, we also obtain that overlap and grouping functions have particular structures with a constant domain where its value equals to the neutral element of the idempotent uninorm when they are distributive over idempotent uninorms.

Fuzzy gyronorms on gyrogroups

The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. In this paper, the notion of fuzzy gyronorms on gyrogroups is introduced. The relations of fuzzy metrics (in the sense of George and Veeramani), fuzzy gyronorms and gyronorms on gyrogroups are studied. Also, the fuzzy metric structures on fuzzy normed gyrogroups are discussed. Finally the fuzzy metric completion of a gyrogroup with an invariant metric is studied. We mainly show that if d is an invariant metric on a gyrogroup G and (Gˆ,dˆ) is the metric completion of the metric space (G,d); then for any continuous t-norm ⁎, the standard fuzzy metric space (Gˆ,Mdˆ,⁎) of (Gˆ,dˆ) is the (up to isometry) unique fuzzy metric completion of the standard fuzzy metric space (G,Md,⁎) of (G,d); furthermore, (Gˆ,Mdˆ,⁎) is a fuzzy metric gyrogroup containing (G,Md,⁎) as a dense fuzzy metric subgyrogroup and Mdˆ is invariant on Gˆ. Applying this result, we obtain that every gyrogroup G with an invariant metric d admits an (up to isometric) unique complete metric space (Gˆ,dˆ) of (G,d) such that Gˆ with the topology introduced by dˆ is a topological gyrogroup containing G as a dense subgyrogroup and dˆ is invariant on Gˆ.

Some New Aspects in the Intuitionistic Fuzzy and Neutrosophic Fixed Point Theory

In this manuscript, we use the concepts of continuous t-norms and continuous t-conorms to introduce some definitions, in which intuitionistic fuzzy rectangular metric spaces, intuitionistic fuzzy rectangular metric-like spaces, intuitionistic fuzzy rectangular b-metric spaces, intuitionistic fuzzy rectangular b-metric-like spaces, neutrosophic rectangular metric spaces, neutrosophic rectangular metric-like spaces, neutrosophic rectangular b-metric spaces, and neutrosophic rectangular b-metric-like spaces are included. Continuous t-norms and continuous t-conorms are used to generalize the probability distribution of triangular inequalities in metric space axioms. Nontrivial examples, some fixed point results, and an application to the integral equation are imparted in this manuscript.

Bipolar fuzzy metric spaces with application

The concept of a bipolar metric space is presented in this article. In bipolar fuzzy metric space, the notions of continuous t-norms and bipolar continuous symmetry t_s-conorms employs important role. Requirements in metric space, triangular norms are used to extrapolate with the probability density function of triangle inequality. Dual processes of triangular norms are known as triangular conorms. Continuous t-norms and continuous symmetry t_s-conorms are used to define bipolar metric space in this paper. Besides, a number of topological and functional properties of the bipolar metric space have been studied. Afterwards, Baire Category and Uniform Convergence Theorems for bipolar metric spaces are presented. After that, an application on determining the appropriate type of vaccine in the treatment process of COVID-19 is given using similarity measure between bipolar fuzzy metric spaces. For vaccine selection and supply chain management, an advanced multi-attribute decision-making (MADM) algorithm is being developed. Finally, the validity of the proposed MADM method for selecting the best proper form of vaccine is also established.

Continuous [0,1]-lattices and injective [0,1]-approach spaces

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective T0 spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in an enriched context, where the enrichment is the quantale obtained by equipping the interval [0,1] with a continuous t-norm. It is shown that for each continuous t-norm, the specialization [0,1]-order of a separated and injective [0,1]-approach space X is a continuous [0,1]-lattice and the [0,1]-approach structure of X coincides with the Scott [0,1]-approach structure of its specialization [0,1]-order; but, unlike in the classical situation, the converse fails in general.