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This paper mainly focuses on new methods of constructing triangular norms as well as triangular conorms on a bounded lattice. At first, we propose the definition of a partition element and show that the definition is well-defined. Then, we construct a t-norm (t-conorm) on a bounded lattice with the help of an interior (a closure) operator and a given t-subnorm (t-subconorm). For the sake of clarity, we give some interpretive examples of the new methods and show that the conditions restricting the elements are indispensable in the construction process. Moreover, we discuss the relationships among the construction methods proposed in this paper and the methods that already exist. Finally, we generalize the new construction methods to modified ordinal sum methods by iteration. [ABSTRACT FROM AUTHOR]

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f (x) = lim n → ∞ x T (n) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) >0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = TM. [ABSTRACT FROM AUTHOR]

In this article, we deal with the algebraic structures of the partial orders induced by non-divisible t-norms. We first give a condition for the partial orders induced by non-divisible t-norms being a meet semi-lattice. Then, we show the condition for the partial orders induced by right-continuous t-norms being a lattice, which is very useful for constructing new lattice structures. Finally, an illustrative example is presented. Our results go a further step towards answering the first open problem raised by Karaçal and Kesicioğlu (Kybernetika 47:300-314, 2011). [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f (x) = lim n → ∞ x T (n) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) >0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = TM. [ABSTRACT FROM AUTHOR]

Uninorms are a special type of associative aggregation functions, which have received widespread attention in the theoretical and practical fields since their introduction. Durante and Sarkoci introduced the migrativity property in 2008. Afterwards, this property was widely applied in numerous fields like image processing and decision analysis, which has sparked a series of studies. There have been a large number of research results on the migrativity involving uninorms, but the work has mainly focused on the uninorms internal on the boundary. In this paper, we will concentrate on the uninorms not internal on the boundary. First, we discuss the characterization of the α-migrativity of conjunctive uninorms over continuous t-norms according to the value of α. Then, the consequences of the α-migrativity of disjunctive uninorms over continuous t-conorms can be obtained dually. [ABSTRACT FROM AUTHOR]

In this study, we introduce the notion of the T-irreducible element as a generalization of the notion of the meet-irreducible element in complete lattices. We derive some related properties of these elements and T-prime elements. We prove that T-irreducible elements and T-prime elements are preserved under the isomorphism that is generated by the same t-norm. We discuss the relationship between the sets of T-prime elements and co-atoms under some conditions. We illustrate this discussion with some examples. We also give some characterizations for the sets of T-irreducible elements and T-prime elements on the direct product of lattices. Then, we show that Theorem 2 given by Karaçal and Sağıroğlu is false by giving some counterexamples. We present a necessary and sufficient condition for the mentioned theorem to be correct. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some fixed point results in generalized Menger probabilistic metric spaces. Moreover, some nontrivial examples are presented to illustrate the superiority of the obtained results. In addition, several interesting applications are given to show that our results are meaningful and valuable. [ABSTRACT FROM AUTHOR]

The fuzzy information system over two universes formalizes a data table corresponding to two universes as well as their relations. Although many analyses have focused on this topic, the existing models are mixed with some uncertain factors in the process of given relation. And it is unnecessary to analyze the fuzzy objects with small membership degree. This motivates us to develop a better method to handle complex data. This paper proposes a local fuzzy rough set model over two universes, then further analyzes its properties and decision rules. Moreover, the reduction method of the model is being studied. The experiment results show that the proposed model has better performance in classfication and decision-making. [ABSTRACT FROM AUTHOR]

As a proper generalization of the ordinal sum t-norm construction on bounded lattices proposed in [E. Aşıcı, R. Mesiar, New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice, International Journal of General Systems, 49(2) (2020), 143-160], the present paper studies a new extension of a triangular norm on a subinterval [0, α] via an interior operator to the underlying entire bounded lattice, where the necessary and sufficient conditions under which the constructed operation is again a t-norm are given. By comparing the graphic structures of two t-norms on a common bounded lattice which are constructed in different ways, it is shown that the new method in this paper is essentially different from the ones existing in the literature. As an end, this new construction is generalized to construct ordinal sums of finitely many t-norms by recursion on bounded lattices. The dual results for ordinal sum construction of t-conorms via closure operators on bounded lattices are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we are mainly to solve the functional equations given by the modularity condition. Modularity condition between disjunctive (resp. conjunctive) uni-nullnorms and some most studied classes of binary aggregation operators (i.e., t-norms, t-conorms, uninorms and semi-t-operators) are discussed. Both positive and negative results of modularity condition for uni-nullnorms are obtained. Since uni-nullnorms are an extension of nullnorms and uninorms, previous results about modularity condition for nullnorms and uninorms in max and min can be obtained as simple corollaries. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Consider a two-place function T : [ 0 , 1 ] 2 → [ 0 , 1 ] defined by T (x , y) = f (− 1) (F (f (x) , f (y))) where F : [ 0 , ∞ ] 2 → [ 0 , ∞ ] is an associative function, f : [ 0 , 1 ] → [ 0 , ∞ ] is a monotone function that satisfies either f (x) = f (x +) when f (x +) ∈ Ran (f) or f (x) ≠ f (y) for any y ≠ x when f (x +) ∉ Ran (f) for all x ∈ [ 0 , 1 ] and f (− 1) : [ 0 , ∞ ] → [ 0 , 1 ] is a pseudo-inverse of f. In this article, the associativity of the function T is shown to depend only on properties of the range of f. The necessary and sufficient conditions for the T being associative are presented by applying the properties of the monotone function f. [ABSTRACT FROM AUTHOR]

This article deals with the formula f (− 1) (F (f (x) , f (y))) generated by a one-place function f : [ 0 , 1 ] → [ 0 , 1 ] and a binary function F : [ 0 , 1 ] 2 → [ 0 , 1 ]. When the f is a strictly increasing function and F is a continuous, non-decreasing and associative function with neutral element in [ 0 , 1 ] , the following algebraic and analytical properties of the formula are studied: idempotent elements, the continuity (resp. left-continuity/right-continuity), the associativity and the limit property. Relationship among these properties is investigated. Some necessary conditions and some sufficient conditions are given for the formula being a triangular norm (resp. triangular conorm). In particular, a necessary and sufficient condition are expressed for obtaining a continuous Archimedean triangular norm (resp. triangular conorm). When the f is a non-decreasing surjective function and F is a non-decreasing associative function with neutral element in [ 0 , 1 ] , we investigate the associativity of the formula. [ABSTRACT FROM AUTHOR]

Fuzzy model checking, also called multi-valued model checking, has proved to be an effective technique in verifying properties of fuzzy systems. One important issue with fuzzy model checking, is that a model adopted in fuzzy model checking is frequently updated with small changes, and it is too costly to run a model-checking algorithm from scratch in response to every update. To address the issue, in this paper, we consider the incremental model-checking approach for fuzzy systems by making maximal use of previous model checking results or in other words, by minimizing unnecessary recomputation. The models of our study are fuzzy Kripke structures, which are a fuzzy counterpart of Kripke structures and used to describe fuzzy systems, while the properties of fuzzy systems are expressed using fuzzy computation tree logic, a fuzzy temporal logic derived from computation tree logic. The focus of the paper is on how to design incremental model-checking algorithms for two until-formulas which characterize the maximal or dually minimum constrained reachability properties with respect to fuzzy Kripke structures under transition insertions or deletions but not both. The feasibility of our approach is illustrated by an example arising from the path planning problem of mobile robots. [ABSTRACT FROM AUTHOR]

The additive generators play an important role in the construction of t-(sub)norms. Distinct to t-norms, the additive generators of t-subnorms need not be strictly decreasing and the value at 1 need not be 0. This paper gives the sufficient and necessary conditions such that two t-subnorms, generated by continuous additive generators, are isomorphic. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

There are many arguments for counting with more than two "truth values"; this allows to imitate human reasoning of facts which are not binary. For theoretical reasons, it is natural to use the whole real interval as the scale. However, this brings practical problems: it is difficult, and even impossible, to represent exact values. Often only a small scale of values suffices to express what we need. Therefore, finite chains are frequently used as domains of fuzzy logical operations. Their representation and manipulation are easy. In this paper, we focus on triangular norms (t-norms). The choice of a finite domain admits some operations (Gödel, Łukasiewicz), while it excludes others (all strict ones, including the product). A disadvantage of the Gödel (minimum) t-norm is that repetition of arguments does not change their meaning. This is often desirable to emphasize the statement. ("Words, words, words!") Thus we do not consider the Gödel operations sufficient for representation of all fuzzy logical statements in human reasoning. Together with them, we discard all operations with idempotent elements other than 0, 1, thus we restrict attention to Archimedean t-norms (and t-conorms as their duals, not treated explicitly in the sequel). A disadvantage of many Archimedean t-norms is that, when applied to several arguments, the result is very often zero. Then, it gives no clue in comparing the outcomes, e.g., in giving priorities to alternatives evaluated by fuzzy rules. For instance, when the continuous Łukasiewicz t-norm is applied to 5 entries, it is nonzero only on 1 / 5 ! = 1 / 120 of the volume of its 5-dimensional domain; in its discrete versions, nonzero results are even more rare. Thus we are interested in Archimedean t-norms with values "as large as possible", here in the maximal Archimedean t-norms (=those which are not majorized by other Archimedean t-norms). It was shown in previous works that there is an abundance of discrete t-norms; their number grows fast with the number of elements of the underlying chain (no exponential bound seems to be known). There is also an abundance of Archimedean t-norms. In contrast to that, when we counted the number of maximal Archimedean t-norms, it grows asymptotically exponentially with a mild base. What is more interesting, these numbers follow the Fibonacci sequence. We have found a description of maximal Archimedean t-norms and explained also the processes behind their construction and the role of the Fibonacci sequence. These results link the t-norms to a construction using a kind of additive generators, although this could not be used without modifications. We do not describe all Archimedean t-norms and their numbers, but we have completed at least a significant step towards this goal, too. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some construction methods for triangular norms and triangular conorms by means of arbitrary elements on any bounded lattices. We examine the structure of these construction methods according to the some special cases of these arbitrary elements on any bounded lattice. On the other hand, we show that these triangular norms (triangular conorms) may not be equal in general. In addition, we provide some illustrative examples for the new constructions of triangular norms and triangular conorms on bounded lattices. [ABSTRACT FROM AUTHOR]

Following a historical overview of the development of ordinal sums, a presentation of the most relevant results for ordinal sums of triangular norms and copulas is given (including gluing of copulas, orthogonal grid constructions and patchwork operators). The ordinal sums of copulas considered here are constructed not only by means of the comonotonic copula, but also by using the lower Fréchet-Hoeffding bound and the independence copula. We provide alternative proofs to some results on ordinal sums, elaborate properties common to all or just some of the ordinal sums discussed. Also included are a discussion of the relationship between ordinal sums of copulas and the Markov product and an overview of ordinal sums of multivariate copulas, illustrating aspects to be considered when extending concepts for ordinal sums of bivariate copulas to the multivariate case. [ABSTRACT FROM AUTHOR]

The paper studies the aggregation of pairs of T -indistinguishability operators. More concretely we address the question whether A (E , E ′) is a T -indistinguishability operator if E , E ′ are T -indistinguishability operators. The answer depends on the aggregation function, the t-norm T , and the chosen T -indistinguishability operators. It is well-known that an aggregation function preserves T -transitive relations if and only if it dominates the t-norm T. We show the important role of the minimum t-norm T M in this preservation problem. In particular we develop weaker forms of domination that are used to provide characterizations of T M -indistinguishability preservation under aggregation. We also prove that the existence of a single strictly monotone aggregation that satisfies the indistinguishability operator preservation property guarantees all aggregations to have the same preservation property. [ABSTRACT FROM AUTHOR]

The study of consistency in decision-making models is of great importance. Our proposal deals with a special type of extended aggregation functions to guarantee consistency that is based on T - S consistency of linearly ordered fuzzy preference relations (fuzzy ranking for preferences). Necessary conditions to ensure consistency in the aggregation process are provided. Moreover, sufficient conditions to guarantee the existence of extended aggregation functions verifying T - S consistency are presented. Some examples of functions of this type are also given. The equivalence between T - S consistency and the additive consistency for fuzzy preference relations is established. [ABSTRACT FROM AUTHOR]

In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying algebraic operations on fuzzy posets and bounded fuzzy lattices. We first prove that fuzzy posets are closed under finite direct products whenever the triangular norm realizing the product construction has no zero divisors. This result is then extended to the case of bounded fuzzy lattices. Some immediate consequences are then obtained within the setting of direct products realized by triangular norms with no nilpotent elements as well as strictly monotone and cancellative triangular norms. We then introduce a triangular norm based construction of ordinal products and similarly show that fuzzy posets are closed under ordinal products whenever the triangular norm realizing the product construction has no zero divisors. [ABSTRACT FROM AUTHOR]

In numerous generalizations of the original theory of fuzzy sets proposed by Zadeh, the considered membership degrees are often taken from lattices isomorphic to the lattice L I of closed subintervals of the unit interval [ 0 , 1 ]. This is, for example, the case of intuitionistic values, Pythagorean values or q -rung orthopair values. The mentioned isomorphisms allow to transfer the results obtained for the lattice L I directly to the other mentioned lattices. In particular, basic connectives in Zadeh's fuzzy set theory, i.e., special functions on the lattice [ 0 , 1 ] , can be extended to the interval-valued connectives, i.e., to special functions on the lattice L I , and then to the connectives on the lattices L ⁎ of intuitionistic values, P of Pythagorean values, and also on the lattice L τ q of q -rung orthopair values. We give several examples of such connectives, in particular, of those which are related to strict t-norms. For all these connectives we show their link to an additive generator f of the considered strict t-norm T. Based on our approach, many results discussed in numerous papers can be treated in a unique framework, and the same is valid for possible newly proposed types of connectives based on strict t-norms. Due to this approach, a lot of tedious proofs of the properties of the proposed connectives could be avoided, which gives researchers more space for presented applications. [ABSTRACT FROM AUTHOR]

First, we present construction methods for interior operators on a meet semilattice. Second, under the assumption that the underlying meet semilattices constitute the range of an interior operator, we prove an ordinal sum theorem for countably many (finite or countably infinite) triangular norms on bounded meet semilattices, which unifies and generalizes two recent results: one by Dvořák and Holčapek and the other by some of the present authors. We also characterize triangular norms that are representable as the ordinal sum of countably many triangular norms on given bounded meet semilattices. [ABSTRACT FROM AUTHOR]

This paper introduces a novel concept of KM-single valued neutrosophic Hausdorff space and KM-single valued neutrosophic manifold space. This study generalizes the concept of KM-single valued neutrosophic manifold space to union and product of KM-single valued neutrosophic manifold space and in this regard investigates some product of KM-single valued neutrosophic manifold spaces. Indeed, this study analyses the notation of KM-single valued neutrosophic manifold based on a valued-level subset. [ABSTRACT FROM AUTHOR]

The ordinal sum of t-norms on a bounded lattice has been used to construct other t-norms. However, an ordinal sum of binary operations (not necessarily t-norms) defined on the fixed subintervals of a bounded lattice may not be a t-norm. Some necessary and sufficient conditions are presented in this article for ensuring that an ordinal sum on a bounded lattice of two binary operations is, in fact, a t-norm. In particular, the results presented here provide an answer to an open problem put forward by Ertuğrul and Yeşilyurt, ordinal sums of triangular norms on bounded lattices, Inf. Sci., vol. 517, (2020) 198–216. [ABSTRACT FROM AUTHOR]

This paper studies non-cooperative games where players are allowed to play their mixed non-additive strategies. Expected payoffs are expressed by so-called fuzzy integrals: Choquet integral, Sugeno integral and generalizations of Sugeno integral obtained by using triangular norms. We consider the existence problem of Nash equilibrium for such games. Positive results for Sugeno integral and its generalizations are obtained. However we provide some example of a game with Choquet payoffs which have no Nash equilibrium. Such example demonstrates that fuzzy integrals based on the maximum operation are more suitable for possibility capacities than Choquet integral which is based on the addition operation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of T-distributive element, factor congruence, simple triangular norm and indecomposable t-norm on a bounded lattice through T-compatible congruence and study their related properties. We observe that the smallest t-norm on Boolean algebra is simple. We obtain a congruence by means of T-distributive element and investigate some properties of this type of congruence. A relation between the direct decomposability of any t-norm T and T-distributive elements whose supremum is T-∨-decomposable is obtained. Further, we show that any t-norm on a bounded lattice is directly indecomposable if and only if its underlying bounded lattice is directly indecomposable. We show that any t-norm T on a finite bounded lattice is isomorphic to a direct product of directly indecomposable t-norms. Finally, some illustrative examples are added. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper continues to study the construction of uni-nullnorms on bounded lattices. At first, we introduce a new method for constructing the smallest uni-nullnorm on an arbitrary bounded lattice L having the elements e , a ∈ L , based on the existence of a uninorm on [ 0 , a ] 2 with the neutral element e and a triangular norm on [ a , 1 ] 2. And then, we propose another new approach to obtain the largest uni-nullnorm on L via a uninorm on [ 0 , a) 2 with the neutral element e and a triangular norm on [ a , 1 ] 2. Furthermore, we provide some corresponding examples to illustrate that our construction methods differ from the existing ones. [ABSTRACT FROM AUTHOR]

We introduce three broad classes of nullnorms on a bounded lattice and lay bare the structure of their members. For that purpose, we introduce particular subsets of a bounded lattice, called upper (resp. lower) beams, and appropriate associative operations on them, called beam (resp. dual beam) operations, which conveniently generalize triangular norms (resp. triangular conorms). It is shown that nullnorms in the first (resp. second) class are characterized by a triangular conorm (resp. triangular norm) and a beam operation (resp. dual beam operation), while nullnorms in the third class are characterized by a triangular conorm and a triangular norm. We also discuss the relationships among these three classes. [ABSTRACT FROM AUTHOR]

As generalizations of the classes Umin and Umax of uninorms on the real unit interval, we introduce the classes Umin and Umax of uninorms on a bounded lattice and lay bare the structure of their members. It is shown that uninorms in Umin (resp. Umax) are characterized by a triangular conorm (resp. triangular norm) and a triangular subnorm (resp. triangular superconorm). The characterization theorems unify all the existing construction methods for these two classes of uninorms known in the literature. We also introduce four new related classes of uninorms on a bounded lattice and present dedicated construction methods using triangular subnorms and triangular superconorms. Finally, connections among the introduced classes of uninorms are discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given an aggregation function, and two fuzzy subgroups of a group, this study investigates whether the aggregation of them is also a fuzzy subgroup. It is proven that if the cardinal of the group is a prime power, then it is always a fuzzy subgroup. For a general group, it is proven that the existence of a binary relation between the fuzzy subgroups determines if their aggregation is a fuzzy subgroup. Moreover, given two fuzzy subgroups, if the aggregation of them is a fuzzy subgroup for some strictly monotone aggregation function, then it is a fuzzy subgroup for any aggregation function. The present study also examines conditions on the aggregation function that characterise the case where the aggregation of two arbitrary fuzzy subgroups is also a fuzzy subgroup. [ABSTRACT FROM AUTHOR]