Continuous Triangular Norms Research Articles

Lax Extensions of Conical I-Semifilter Monads

For a quantale I, the unit interval endowed with a continuous triangular norm, we introduce the canonical, op-canonical and Kleisli extensions of the conical I-semifilter monad to I-Rel. It is proved that the op-canonical extension coincides with the Kleisli extension.

Sugeno Integral Based on Overlap Function and Its Application to Fuzzy Quantifiers and Multi-Attribute Decision-Making

The overlap function is an important class of aggregation function that is closely related to the continuous triangular norm. It has important applications in information fusion, image processing, information classification, intelligent decision-making, etc. The usual multi-attribute decision-making (MADM) is to select the decision object that performs well on all attributes (indicators), which is quite demanding. The MADM based on fuzzy quantifiers is to select the decision object that performs well on a certain proportion or quantification (such as most, many, more than half, etc.) of attributes. Therefore, it is necessary to study how to express and calculate fuzzy quantifiers such as most, many, etc. In this paper, the Sugeno integral based on the overlap function (called the O-Sugeno integral) is used as a new information fusion tool, and some related properties are studied. Then, the truth value of a linguistic quantified proposition can be estimated by using the O-Sugeno integral, and the O-Sugeno integral semantics of fuzzy quantifiers is proposed. Finally, the MADM method based on the O-Sugeno integral semantics of fuzzy quantifiers is proposed and the feasibility of our method is verified by several illustrative examples such as the logistics park location problem.

Spaces of non-additive measures generated by triangular norms

We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.

On Discrete Presheaf Monads

For a quantale I, which is a unit interval endowed with a continuous triangular norm and the Barr extension β¯I of the ultrafilter monad to I-Rel, a characterization of the discrete presheaf monad associated to β¯I is given. It is also proved that, when & is the Łucasiewicz triangular norm, the discrete presheaf monad is isomorphic to the saturated prefilter monad, and when & is the product triangular norm, the prime functional ideal monad is isomorphic to a submonad of the discrete presheaf monad.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

In this manuscript, our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces. We establish some fixed point theorems in this setting. Also, we plot some graphs of an example of obtained result for better understanding. We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.

Fuzzy number-valued triangular norm-based decomposable time-stamped fuzzy measure and integration

In this paper, we introduce a new concept of fuzzy measure and fuzzy integration which has a dynamic position and is different from previous approaches. Our definition of a new type of fuzzy measure deals with distance functions (special L-fuzzy numbers) and is based on continuous triangular norms. By this concept, we construct a new version of measure theory and integration which is more flexible since the measure of the set both depends on the set itself and on the other parameter named by time. Our approach is related to the idea of fuzzy metric spaces. We study some fuzzy measures induced by classical measures. An integral based on the introduced measures is proposed and studied, too. To complete our paper, we prove some limits and convergence theorems.

Convex combinations of uninorms and triangular subnorms

The problem whether a non-trivial convex combination of two continuous triangular norms can be a t-norm was extensively studied in the last decades, bringing the negative answer in all examined special cases. On the other hand, in the field of non-continuous t-norms (including left-continuous t-norms) several examples were found, including mainly a convex combination of an α-migrative t-norm and the drastic product t-norm, or a convex combination of ordinal sums of product related t-subnorms. Recently, convex combinations of uninorms were discussed in [18]. Since any uninorm with the neutral element e∈[0,1] restricted to [0,e]2 is isomorphic to a t-norm, this investigation extends the original problem of a convex combination of t-norms given by Alsina, Frank and Schweizer. In this work we will study convex combinations of uninorms further, covering uninorms with continuous underlying functions, uninorms with different neutral elements and representable uninorms. We will also show that convex combinations of triangular subnorms play a major role in this investigation.

Neutrosophic metric spaces

Neutrosophy consists of neutrosophic logic, probability, and sets. Actually, the neutrosophic set is a generalisation of classical sets, fuzzy set, intuitionistic fuzzy set, etc. A neutrosophic set is a mathematical notion serving issues containing inconsistent, indeterminate, and imprecise data. The notion of intuitionistic fuzzy metric space is useful in modelling some phenomena where it is necessary to study the relationship between two probability functions. In this paper, the definition of new metric space with neutrosophic numbers is given. Neutrosophic metric space uses the idea of continuous triangular norms and continuous triangular conorms in intuitionistic fuzzy metric space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. Triangular norms and triangular conorm are very significant for fuzzy operations. Neutrosophic metric space was defined with continuous triangular norms and continuous triangular conorms. Several topological and structural properties neutrosophic metric space have been investigated. The analogues of Baire Category Theorem and Uniform Convergence Theorem are given for Neutrosophic metric spaces.

On the construction of fuzzy betweenness relations from metrics

We consider the problem of constructing a fuzzy betweenness relation from a metric. More precisely, given a continuous Archimedean triangular norm, we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo-inverse of either a continuous additive generator or a continuous multiplicative generator of the triangular norm. In case the metric is bounded and given a 1-Lipschitz continuous triangular norm, we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm. Since the Łukasiewicz and product triangular norms are both continuous Archimedean and 1-Lipschitz continuous, all three construction methods may be used. Interestingly, the construction method based on the residual implication is proved to coincide with that based on a continuous additive generator for the Łukasiewicz triangular norm and with that based on a continuous multiplicative generator for the product triangular norm. We end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric.

Q-Rung Orthopair Fuzzy Integrals in the Frame of Continuous Archimedean T-Norms and T-Conorms and Their Application

Yager's q-rung orthopair fuzzy set is a generalization of fuzzy sets, whose prominent feature is that the qth power sum of the membership and the nonmembership degrees is equal to or less than one, and we call its core, an ordered pair, q-rung orthopair fuzzy number (q-ROFN). More recently, the scholars have constructed the q-rung orthopair fuzzy calculus (q-ROFC), which can effectively deal with continuous q-rung orthopair fuzzy information. Nevertheless, the q-ROFC is only based on the basic operational laws of the q-ROFNs, in fuzzy theory, Archimedean t-norms and t-conorms (ATTs) are a significant class of continuous triangular norms and conorms, which are the generalizations of the intersection and union related to fuzzy sets. Thus, in order to extend the q-ROFC to a wider area, in this article, we systematically discuss the q-rung orthopair fuzzy double integrals (q-ROFDIs) in the frame of ATTs. First, we construct the q-ROFDI in the frame of Archimedean t-conorms in detail, and then provide its concrete value. In addition, we reveal the relationships with respect to two types of q-rung orthopair fuzzy spaces. Based on which, we can easily obtain another types of q-ROFDI. After that, we investigate their fundamental properties in detail so as to comprehend these kinds of q-ROFDIs in-depth. Finally, we point out the essences of these kinds of q-ROFDIs, on the basis of which we provide a practical application to show their effectiveness and elasticity via comparing with the existing methods.

On the distributivity of continuous triangular norms and triangular conorms with respect to 2-uninorms

In this paper two results of the functional equation of distributivity for 2-uninorms published by Drygaś and Rak in [4] are revisited. In particular, we present some counterexamples to considerations covered by Theorems 4.3 and 4.4 in the mentioned paper, and then we correct them accordingly.

Continuous triangular norm based fuzzy topology

For each continuous t-norm &, a class of fuzzy topological spaces, called &-topological spaces, is introduced. The motivation stems from the idea that to each many-valued logic there may correspond a theory of many-valued topology, in particular, each continuous t-norm may lead to a theory of fuzzy topology. It is shown that for each continuous t-norm &, the subcategory consisting of &-topological spaces is simultaneously reflective and coreflective in the category of fuzzy topological spaces, hence gives rise to an autonomous theory of fuzzy topology. Topologizing a fuzzy pre-ordered set with the fuzzy Scott topology yields a functor from the category of fuzzy pre-ordered sets and maps that preserve suprema of flat ideals to the category of &-topological spaces. It is proved that this functor is a full one if and only if the t-norm & is Archimedean.

The cross-migrativity with respect to continuous triangular norms revisited

The structures of continuous cross-migrative triangular norms with respective to some continuous triangular norm have been studied, and only partial results were obtained. This paper is devoted to presenting a complete characterization of the class of continuous triangular norms, being cross-migrative with respect to some continuous triangular norm.

On (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices

Rough sets were introduced by Pawlak as a formal tool for dealing with imprecision and uncertainty in data analysis. After that time, varieties of fuzzy generalizations of rough approximations have been investigated in the literature. In particular, generalized fuzzy rough sets determined by a triangular norm have been considered. On the other hand, binary operations & and ⊙ on complete residuated and co-residuated lattice L can be regarded as two more extensive operations than left continuous triangular norms and right continuous triangular conorms, respectively. Therefore, as a further generalization of the notion of rough sets, this paper is devoted to proposing (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices from both constructive and axiomatic approaches. In the constructive approach, we define a pair of lower and upper L-fuzzy rough approximation operators determined by ⊙ and &, respectively. Meanwhile, various classes of (⊙,&)-fuzzy rough sets are investigated. In the axiomatic approach, the axiomatic characterizations of different (⊙,&)-fuzzy rough approximation operators are studied. Furthermore, the topological properties of (⊙,&)-fuzzy rough sets are discussed, especially from the categorical point of view. At the end of this paper, we give a brief introduction to a new model of fuzzy rough sets and discuss some properties of it.

Principles of inclusion and exclusion for interval-valued fuzzy sets and IF-sets

The principle of inclusion and exclusion is one of basic tools of classical set theory. Many previous papers have presented its generalizations to fuzzy sets or IF-sets (Atanassov's intuitionistic fuzzy sets) with Gödel or product operations. In our previous work we clarified these questions for fuzzy sets with intersections and unions based on continuous triangular norms and conorms. Here we extend these results to interval-valued fuzzy sets and IF-sets.

Yoneda completeness and flat completeness of ordered fuzzy sets

This paper studies Yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm. Both of these notions are natural extension of directed completeness in order theory to the fuzzy setting. Yoneda completeness requires every forward Cauchy net converges (has a Yoneda limit), while flat completeness requires every flat weight (a counterpart of ideals in partially ordered sets) has a supremum. It is proved that flat completeness implies Yoneda completeness, but, the converse implication holds only in the case that the related triangular norm is either isomorphic to the Łukasiewicz t-norm or to the product t-norm.

Finitely maxitive T-conditional possibility theory: Coherence and extension

Starting from the axiomatic definition of finitely maxitive T-conditional possibility (where T is a continuous triangular norm), the paper aims at a comprehensive and self-contained treatment of coherence and extension of a possibilistic assessment defined on an arbitrary set of conditional events. Coherence (or consistence with a T-conditional possibility) is characterized either in terms of existence of a linearly ordered class of finitely maxitive possibility measures (T-nested class) agreeing with the assessment, or in terms of solvability of a finite sequence of nonlinear systems for every finite subfamily of conditional events. Coherence reveals to be a necessary and sufficient condition for the extendibility of an assessment to any superset of conditional events and, in the case of T equal to the minimum or a strict t-norm, the set of coherent values for the possibility of a new conditional event can be computed solving two optimization problems over a finite sequence of nonlinear systems for every finite subfamily of conditional events.

Convex combinations of fuzzy logical operations

One may ask when (non-trivial) convex combinations of fuzzy logical operations result in operations of the same type. This question has been intensively studied for triangular norms and it still remains open for continuous ones. An equivalent problem is obtained for triangular conorms. We show equivalence with the analogous problem for S-implications. Negative answers are obtained for strong fuzzy negations and for R-implications corresponding to continuous triangular norms. The situation for Q-implications and related problems are discussed.

Determinization of fuzzy automata via factorization of fuzzy states

Fuzzy finite-state automata over the algebra ([0,1],max,⊗,0,1), in which the monoid ([0,1],⊗,1) (⊗ denotes a continuous triangular norm) is not locally finite, can accept fuzzy languages of infinite range. For a given fuzzy finite-state automaton which accepts a fuzzy language of infinite range, we define the determinization of the fuzzy automaton via factorization of fuzzy states, i.e., the computation of an equivalent deterministic fuzzy automaton whether it is finite. This method of determinization is a generalization of the well-known accessible subset construction. Our main contribution is to determine that the representable-cycles property is the necessary and sufficient condition for determinization of a fuzzy finite-state automaton via a maximal factorization of fuzzy states. This property is more general than the twins property (adapted for fuzzy automata) which is the sufficient condition for weighted automata over the tropical semiring.

Fuzzy languages with infinite range accepted by fuzzy automata: Pumping Lemma and determinization procedure

The formulation of fuzzy automata allows us to select a great variety of triangular norms. Depending on the selected triangular norm, a fuzzy automaton can accept a fuzzy language (FA-language) with infinite range. These fuzzy automata are not equivalent to the so-called deterministic fuzzy automata (deterministic automata with a fuzzy subset of final states) which only accept fuzzy languages with finite range. In this paper, we study FA-languages with infinite range and a determinization procedure in order to obtain an equivalent fuzzy deterministic automaton for a given fuzzy automaton. A fuzzy deterministic automaton is a fuzzy automaton which satisfies the deterministic condition in its state transition function. The main contributions of our paper are: (1) a Pumping Lemma of FA-languages with infinite range; (2) the formulation of fuzzy deterministic automata and a Pumping Lemma of FDA-languages; (3) the necessary conditions for the determinization of fuzzy automata under continuous triangular norms which accept fuzzy languages of infinite range; and (4) a determinization algorithm for fuzzy automata, its correctness proof and performance.